# Properties

 Label 693.2.a.b Level $693$ Weight $2$ Character orbit 693.a Self dual yes Analytic conductor $5.534$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(1,693)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 693.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$5.53363286007$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{4} - 3 q^{5} + q^{7}+O(q^{10})$$ q - 2 * q^4 - 3 * q^5 + q^7 $$q - 2 q^{4} - 3 q^{5} + q^{7} + q^{11} - 4 q^{13} + 4 q^{16} + 6 q^{17} + 2 q^{19} + 6 q^{20} - 3 q^{23} + 4 q^{25} - 2 q^{28} + 6 q^{29} + 5 q^{31} - 3 q^{35} + 11 q^{37} - 6 q^{41} + 8 q^{43} - 2 q^{44} + q^{49} + 8 q^{52} + 6 q^{53} - 3 q^{55} + 9 q^{59} - 10 q^{61} - 8 q^{64} + 12 q^{65} + 5 q^{67} - 12 q^{68} - 9 q^{71} + 2 q^{73} - 4 q^{76} + q^{77} - 10 q^{79} - 12 q^{80} - 12 q^{83} - 18 q^{85} + 3 q^{89} - 4 q^{91} + 6 q^{92} - 6 q^{95} - q^{97}+O(q^{100})$$ q - 2 * q^4 - 3 * q^5 + q^7 + q^11 - 4 * q^13 + 4 * q^16 + 6 * q^17 + 2 * q^19 + 6 * q^20 - 3 * q^23 + 4 * q^25 - 2 * q^28 + 6 * q^29 + 5 * q^31 - 3 * q^35 + 11 * q^37 - 6 * q^41 + 8 * q^43 - 2 * q^44 + q^49 + 8 * q^52 + 6 * q^53 - 3 * q^55 + 9 * q^59 - 10 * q^61 - 8 * q^64 + 12 * q^65 + 5 * q^67 - 12 * q^68 - 9 * q^71 + 2 * q^73 - 4 * q^76 + q^77 - 10 * q^79 - 12 * q^80 - 12 * q^83 - 18 * q^85 + 3 * q^89 - 4 * q^91 + 6 * q^92 - 6 * q^95 - q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 −2.00000 −3.00000 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.a.b 1
3.b odd 2 1 77.2.a.b 1
7.b odd 2 1 4851.2.a.k 1
11.b odd 2 1 7623.2.a.i 1
12.b even 2 1 1232.2.a.d 1
15.d odd 2 1 1925.2.a.f 1
15.e even 4 2 1925.2.b.g 2
21.c even 2 1 539.2.a.b 1
21.g even 6 2 539.2.e.e 2
21.h odd 6 2 539.2.e.d 2
24.f even 2 1 4928.2.a.x 1
24.h odd 2 1 4928.2.a.i 1
33.d even 2 1 847.2.a.c 1
33.f even 10 4 847.2.f.g 4
33.h odd 10 4 847.2.f.f 4
84.h odd 2 1 8624.2.a.s 1
231.h odd 2 1 5929.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.b 1 3.b odd 2 1
539.2.a.b 1 21.c even 2 1
539.2.e.d 2 21.h odd 6 2
539.2.e.e 2 21.g even 6 2
693.2.a.b 1 1.a even 1 1 trivial
847.2.a.c 1 33.d even 2 1
847.2.f.f 4 33.h odd 10 4
847.2.f.g 4 33.f even 10 4
1232.2.a.d 1 12.b even 2 1
1925.2.a.f 1 15.d odd 2 1
1925.2.b.g 2 15.e even 4 2
4851.2.a.k 1 7.b odd 2 1
4928.2.a.i 1 24.h odd 2 1
4928.2.a.x 1 24.f even 2 1
5929.2.a.d 1 231.h odd 2 1
7623.2.a.i 1 11.b odd 2 1
8624.2.a.s 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(693))$$:

 $$T_{2}$$ T2 $$T_{5} + 3$$ T5 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T - 1$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T - 6$$
$19$ $$T - 2$$
$23$ $$T + 3$$
$29$ $$T - 6$$
$31$ $$T - 5$$
$37$ $$T - 11$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T - 9$$
$61$ $$T + 10$$
$67$ $$T - 5$$
$71$ $$T + 9$$
$73$ $$T - 2$$
$79$ $$T + 10$$
$83$ $$T + 12$$
$89$ $$T - 3$$
$97$ $$T + 1$$