# Properties

 Label 693.2.a.h Level $693$ Weight $2$ Character orbit 693.a Self dual yes Analytic conductor $5.534$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 3 q^{4} + 2 q^{5} + q^{7} - \beta q^{8} +O(q^{10})$$ q - b * q^2 + 3 * q^4 + 2 * q^5 + q^7 - b * q^8 $$q - \beta q^{2} + 3 q^{4} + 2 q^{5} + q^{7} - \beta q^{8} - 2 \beta q^{10} + q^{11} + (\beta + 1) q^{13} - \beta q^{14} - q^{16} + (\beta + 1) q^{17} + (2 \beta + 2) q^{19} + 6 q^{20} - \beta q^{22} + ( - 2 \beta + 2) q^{23} - q^{25} + ( - \beta - 5) q^{26} + 3 q^{28} + ( - 2 \beta - 4) q^{29} + (\beta - 5) q^{31} + 3 \beta q^{32} + ( - \beta - 5) q^{34} + 2 q^{35} + ( - 2 \beta - 4) q^{37} + ( - 2 \beta - 10) q^{38} - 2 \beta q^{40} + (\beta + 9) q^{41} + 8 q^{43} + 3 q^{44} + ( - 2 \beta + 10) q^{46} + (\beta - 5) q^{47} + q^{49} + \beta q^{50} + (3 \beta + 3) q^{52} + (2 \beta - 4) q^{53} + 2 q^{55} - \beta q^{56} + (4 \beta + 10) q^{58} + (\beta - 1) q^{59} + ( - \beta - 5) q^{61} + (5 \beta - 5) q^{62} - 13 q^{64} + (2 \beta + 2) q^{65} + (2 \beta + 10) q^{67} + (3 \beta + 3) q^{68} - 2 \beta q^{70} + (2 \beta + 6) q^{71} + (\beta - 3) q^{73} + (4 \beta + 10) q^{74} + (6 \beta + 6) q^{76} + q^{77} - 4 \beta q^{79} - 2 q^{80} + ( - 9 \beta - 5) q^{82} + (6 \beta - 2) q^{83} + (2 \beta + 2) q^{85} - 8 \beta q^{86} - \beta q^{88} - 2 q^{89} + (\beta + 1) q^{91} + ( - 6 \beta + 6) q^{92} + (5 \beta - 5) q^{94} + (4 \beta + 4) q^{95} + (6 \beta + 4) q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + 3 * q^4 + 2 * q^5 + q^7 - b * q^8 - 2*b * q^10 + q^11 + (b + 1) * q^13 - b * q^14 - q^16 + (b + 1) * q^17 + (2*b + 2) * q^19 + 6 * q^20 - b * q^22 + (-2*b + 2) * q^23 - q^25 + (-b - 5) * q^26 + 3 * q^28 + (-2*b - 4) * q^29 + (b - 5) * q^31 + 3*b * q^32 + (-b - 5) * q^34 + 2 * q^35 + (-2*b - 4) * q^37 + (-2*b - 10) * q^38 - 2*b * q^40 + (b + 9) * q^41 + 8 * q^43 + 3 * q^44 + (-2*b + 10) * q^46 + (b - 5) * q^47 + q^49 + b * q^50 + (3*b + 3) * q^52 + (2*b - 4) * q^53 + 2 * q^55 - b * q^56 + (4*b + 10) * q^58 + (b - 1) * q^59 + (-b - 5) * q^61 + (5*b - 5) * q^62 - 13 * q^64 + (2*b + 2) * q^65 + (2*b + 10) * q^67 + (3*b + 3) * q^68 - 2*b * q^70 + (2*b + 6) * q^71 + (b - 3) * q^73 + (4*b + 10) * q^74 + (6*b + 6) * q^76 + q^77 - 4*b * q^79 - 2 * q^80 + (-9*b - 5) * q^82 + (6*b - 2) * q^83 + (2*b + 2) * q^85 - 8*b * q^86 - b * q^88 - 2 * q^89 + (b + 1) * q^91 + (-6*b + 6) * q^92 + (5*b - 5) * q^94 + (4*b + 4) * q^95 + (6*b + 4) * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{4} + 4 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 6 * q^4 + 4 * q^5 + 2 * q^7 $$2 q + 6 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 2 q^{16} + 2 q^{17} + 4 q^{19} + 12 q^{20} + 4 q^{23} - 2 q^{25} - 10 q^{26} + 6 q^{28} - 8 q^{29} - 10 q^{31} - 10 q^{34} + 4 q^{35} - 8 q^{37} - 20 q^{38} + 18 q^{41} + 16 q^{43} + 6 q^{44} + 20 q^{46} - 10 q^{47} + 2 q^{49} + 6 q^{52} - 8 q^{53} + 4 q^{55} + 20 q^{58} - 2 q^{59} - 10 q^{61} - 10 q^{62} - 26 q^{64} + 4 q^{65} + 20 q^{67} + 6 q^{68} + 12 q^{71} - 6 q^{73} + 20 q^{74} + 12 q^{76} + 2 q^{77} - 4 q^{80} - 10 q^{82} - 4 q^{83} + 4 q^{85} - 4 q^{89} + 2 q^{91} + 12 q^{92} - 10 q^{94} + 8 q^{95} + 8 q^{97}+O(q^{100})$$ 2 * q + 6 * q^4 + 4 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 - 2 * q^16 + 2 * q^17 + 4 * q^19 + 12 * q^20 + 4 * q^23 - 2 * q^25 - 10 * q^26 + 6 * q^28 - 8 * q^29 - 10 * q^31 - 10 * q^34 + 4 * q^35 - 8 * q^37 - 20 * q^38 + 18 * q^41 + 16 * q^43 + 6 * q^44 + 20 * q^46 - 10 * q^47 + 2 * q^49 + 6 * q^52 - 8 * q^53 + 4 * q^55 + 20 * q^58 - 2 * q^59 - 10 * q^61 - 10 * q^62 - 26 * q^64 + 4 * q^65 + 20 * q^67 + 6 * q^68 + 12 * q^71 - 6 * q^73 + 20 * q^74 + 12 * q^76 + 2 * q^77 - 4 * q^80 - 10 * q^82 - 4 * q^83 + 4 * q^85 - 4 * q^89 + 2 * q^91 + 12 * q^92 - 10 * q^94 + 8 * q^95 + 8 * 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
 1.61803 −0.618034
−2.23607 0 3.00000 2.00000 0 1.00000 −2.23607 0 −4.47214
1.2 2.23607 0 3.00000 2.00000 0 1.00000 2.23607 0 4.47214
 $$n$$: 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.h 2
3.b odd 2 1 77.2.a.d 2
7.b odd 2 1 4851.2.a.y 2
11.b odd 2 1 7623.2.a.bl 2
12.b even 2 1 1232.2.a.m 2
15.d odd 2 1 1925.2.a.r 2
15.e even 4 2 1925.2.b.h 4
21.c even 2 1 539.2.a.f 2
21.g even 6 2 539.2.e.j 4
21.h odd 6 2 539.2.e.i 4
24.f even 2 1 4928.2.a.bv 2
24.h odd 2 1 4928.2.a.bm 2
33.d even 2 1 847.2.a.f 2
33.f even 10 2 847.2.f.b 4
33.f even 10 2 847.2.f.m 4
33.h odd 10 2 847.2.f.a 4
33.h odd 10 2 847.2.f.n 4
84.h odd 2 1 8624.2.a.ce 2
231.h odd 2 1 5929.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 3.b odd 2 1
539.2.a.f 2 21.c even 2 1
539.2.e.i 4 21.h odd 6 2
539.2.e.j 4 21.g even 6 2
693.2.a.h 2 1.a even 1 1 trivial
847.2.a.f 2 33.d even 2 1
847.2.f.a 4 33.h odd 10 2
847.2.f.b 4 33.f even 10 2
847.2.f.m 4 33.f even 10 2
847.2.f.n 4 33.h odd 10 2
1232.2.a.m 2 12.b even 2 1
1925.2.a.r 2 15.d odd 2 1
1925.2.b.h 4 15.e even 4 2
4851.2.a.y 2 7.b odd 2 1
4928.2.a.bm 2 24.h odd 2 1
4928.2.a.bv 2 24.f even 2 1
5929.2.a.m 2 231.h odd 2 1
7623.2.a.bl 2 11.b odd 2 1
8624.2.a.ce 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(693))$$:

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{5} - 2$$ T5 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$T^{2}$$
$5$ $$(T - 2)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$T^{2} - 2T - 4$$
$19$ $$T^{2} - 4T - 16$$
$23$ $$T^{2} - 4T - 16$$
$29$ $$T^{2} + 8T - 4$$
$31$ $$T^{2} + 10T + 20$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} - 18T + 76$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} + 10T + 20$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} + 2T - 4$$
$61$ $$T^{2} + 10T + 20$$
$67$ $$T^{2} - 20T + 80$$
$71$ $$T^{2} - 12T + 16$$
$73$ $$T^{2} + 6T + 4$$
$79$ $$T^{2} - 80$$
$83$ $$T^{2} + 4T - 176$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} - 8T - 164$$