# Properties

 Label 693.2.m.k Level 693 Weight 2 Character orbit 693.m Analytic conductor 5.534 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 693.m (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.53363286007$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{5})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 10q^{4} - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 10q^{4} - 8q^{7} - 12q^{10} + 10q^{13} - 26q^{16} - 10q^{19} - 6q^{22} - 52q^{25} - 10q^{28} - 20q^{31} + 24q^{34} - 4q^{37} + 124q^{40} + 32q^{43} + 30q^{46} - 8q^{49} + 24q^{52} - 12q^{55} - 104q^{58} + 34q^{61} - 52q^{64} + 104q^{67} + 18q^{70} + 2q^{73} - 28q^{76} - 42q^{79} - 172q^{82} - 30q^{85} - 16q^{88} - 10q^{91} + 150q^{94} - 74q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1 −0.804651 + 2.47646i 0 −3.86736 2.80980i 1.33452 + 4.10723i 0 −0.809017 0.587785i 5.85703 4.25538i 0 −11.2452
64.2 −0.643412 + 1.98022i 0 −1.88925 1.37262i −0.411680 1.26702i 0 −0.809017 0.587785i 0.564713 0.410288i 0 2.77386
64.3 −0.405703 + 1.24862i 0 0.223566 + 0.162430i −1.06900 3.29005i 0 −0.809017 0.587785i −2.41780 + 1.75664i 0 4.54174
64.4 −0.123907 + 0.381345i 0 1.48796 + 1.08107i 0.712476 + 2.19277i 0 −0.809017 0.587785i −1.24541 + 0.904845i 0 −0.924485
64.5 0.123907 0.381345i 0 1.48796 + 1.08107i −0.712476 2.19277i 0 −0.809017 0.587785i 1.24541 0.904845i 0 −0.924485
64.6 0.405703 1.24862i 0 0.223566 + 0.162430i 1.06900 + 3.29005i 0 −0.809017 0.587785i 2.41780 1.75664i 0 4.54174
64.7 0.643412 1.98022i 0 −1.88925 1.37262i 0.411680 + 1.26702i 0 −0.809017 0.587785i −0.564713 + 0.410288i 0 2.77386
64.8 0.804651 2.47646i 0 −3.86736 2.80980i −1.33452 4.10723i 0 −0.809017 0.587785i −5.85703 + 4.25538i 0 −11.2452
190.1 −2.16885 + 1.57576i 0 1.60284 4.93305i 0.946867 + 0.687939i 0 0.309017 0.951057i 2.64012 + 8.12545i 0 −3.13763
190.2 −1.70216 + 1.23669i 0 0.749912 2.30799i −3.03419 2.20447i 0 0.309017 0.951057i 0.277470 + 0.853964i 0 7.89093
190.3 −0.727975 + 0.528905i 0 −0.367826 + 1.13205i 0.650418 + 0.472557i 0 0.309017 0.951057i −0.887104 2.73023i 0 −0.723426
190.4 −0.614338 + 0.446343i 0 −0.439845 + 1.35370i 2.31804 + 1.68415i 0 0.309017 0.951057i −0.803314 2.47235i 0 −2.17577
190.5 0.614338 0.446343i 0 −0.439845 + 1.35370i −2.31804 1.68415i 0 0.309017 0.951057i 0.803314 + 2.47235i 0 −2.17577
190.6 0.727975 0.528905i 0 −0.367826 + 1.13205i −0.650418 0.472557i 0 0.309017 0.951057i 0.887104 + 2.73023i 0 −0.723426
190.7 1.70216 1.23669i 0 0.749912 2.30799i 3.03419 + 2.20447i 0 0.309017 0.951057i −0.277470 0.853964i 0 7.89093
190.8 2.16885 1.57576i 0 1.60284 4.93305i −0.946867 0.687939i 0 0.309017 0.951057i −2.64012 8.12545i 0 −3.13763
379.1 −0.804651 2.47646i 0 −3.86736 + 2.80980i 1.33452 4.10723i 0 −0.809017 + 0.587785i 5.85703 + 4.25538i 0 −11.2452
379.2 −0.643412 1.98022i 0 −1.88925 + 1.37262i −0.411680 + 1.26702i 0 −0.809017 + 0.587785i 0.564713 + 0.410288i 0 2.77386
379.3 −0.405703 1.24862i 0 0.223566 0.162430i −1.06900 + 3.29005i 0 −0.809017 + 0.587785i −2.41780 1.75664i 0 4.54174
379.4 −0.123907 0.381345i 0 1.48796 1.08107i 0.712476 2.19277i 0 −0.809017 + 0.587785i −1.24541 0.904845i 0 −0.924485
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 631.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.m.k 32
3.b odd 2 1 inner 693.2.m.k 32
11.c even 5 1 inner 693.2.m.k 32
11.c even 5 1 7623.2.a.dc 16
11.d odd 10 1 7623.2.a.db 16
33.f even 10 1 7623.2.a.db 16
33.h odd 10 1 inner 693.2.m.k 32
33.h odd 10 1 7623.2.a.dc 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.m.k 32 1.a even 1 1 trivial
693.2.m.k 32 3.b odd 2 1 inner
693.2.m.k 32 11.c even 5 1 inner
693.2.m.k 32 33.h odd 10 1 inner
7623.2.a.db 16 11.d odd 10 1
7623.2.a.db 16 33.f even 10 1
7623.2.a.dc 16 11.c even 5 1
7623.2.a.dc 16 33.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(693, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database