# Properties

 Label 693.2.i.a Level $693$ Weight $2$ Character orbit 693.i Analytic conductor $5.534$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(100,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("693.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## 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: no Analytic conductor: $$5.53363286007$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{4} + (\zeta_{6} + 2) q^{7} +O(q^{10})$$ q - 2*z * q^2 + (2*z - 2) * q^4 + (z + 2) * q^7 $$q - 2 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{4} + (\zeta_{6} + 2) q^{7} + (\zeta_{6} - 1) q^{11} - 5 q^{13} + ( - 6 \zeta_{6} + 2) q^{14} + 4 \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} - 7 \zeta_{6} q^{19} + 2 q^{22} - 4 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + 10 \zeta_{6} q^{26} + (4 \zeta_{6} - 6) q^{28} + 2 q^{29} + ( - 7 \zeta_{6} + 7) q^{31} + ( - 8 \zeta_{6} + 8) q^{32} - 12 q^{34} - 7 \zeta_{6} q^{37} + (14 \zeta_{6} - 14) q^{38} - 4 q^{41} - 9 q^{43} - 2 \zeta_{6} q^{44} + (8 \zeta_{6} - 8) q^{46} + 6 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} - 10 q^{50} + ( - 10 \zeta_{6} + 10) q^{52} + (2 \zeta_{6} - 2) q^{53} - 4 \zeta_{6} q^{58} + ( - 12 \zeta_{6} + 12) q^{59} + 2 \zeta_{6} q^{61} - 14 q^{62} - 8 q^{64} + (7 \zeta_{6} - 7) q^{67} + 12 \zeta_{6} q^{68} - 8 q^{71} + ( - 5 \zeta_{6} + 5) q^{73} + (14 \zeta_{6} - 14) q^{74} + 14 q^{76} + (2 \zeta_{6} - 3) q^{77} + 11 \zeta_{6} q^{79} + 8 \zeta_{6} q^{82} + 4 q^{83} + 18 \zeta_{6} q^{86} + 6 \zeta_{6} q^{89} + ( - 5 \zeta_{6} - 10) q^{91} + 8 q^{92} + ( - 12 \zeta_{6} + 12) q^{94} + 2 q^{97} + ( - 16 \zeta_{6} + 10) q^{98} +O(q^{100})$$ q - 2*z * q^2 + (2*z - 2) * q^4 + (z + 2) * q^7 + (z - 1) * q^11 - 5 * q^13 + (-6*z + 2) * q^14 + 4*z * q^16 + (-6*z + 6) * q^17 - 7*z * q^19 + 2 * q^22 - 4*z * q^23 + (-5*z + 5) * q^25 + 10*z * q^26 + (4*z - 6) * q^28 + 2 * q^29 + (-7*z + 7) * q^31 + (-8*z + 8) * q^32 - 12 * q^34 - 7*z * q^37 + (14*z - 14) * q^38 - 4 * q^41 - 9 * q^43 - 2*z * q^44 + (8*z - 8) * q^46 + 6*z * q^47 + (5*z + 3) * q^49 - 10 * q^50 + (-10*z + 10) * q^52 + (2*z - 2) * q^53 - 4*z * q^58 + (-12*z + 12) * q^59 + 2*z * q^61 - 14 * q^62 - 8 * q^64 + (7*z - 7) * q^67 + 12*z * q^68 - 8 * q^71 + (-5*z + 5) * q^73 + (14*z - 14) * q^74 + 14 * q^76 + (2*z - 3) * q^77 + 11*z * q^79 + 8*z * q^82 + 4 * q^83 + 18*z * q^86 + 6*z * q^89 + (-5*z - 10) * q^91 + 8 * q^92 + (-12*z + 12) * q^94 + 2 * q^97 + (-16*z + 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{4} + 5 q^{7}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^4 + 5 * q^7 $$2 q - 2 q^{2} - 2 q^{4} + 5 q^{7} - q^{11} - 10 q^{13} - 2 q^{14} + 4 q^{16} + 6 q^{17} - 7 q^{19} + 4 q^{22} - 4 q^{23} + 5 q^{25} + 10 q^{26} - 8 q^{28} + 4 q^{29} + 7 q^{31} + 8 q^{32} - 24 q^{34} - 7 q^{37} - 14 q^{38} - 8 q^{41} - 18 q^{43} - 2 q^{44} - 8 q^{46} + 6 q^{47} + 11 q^{49} - 20 q^{50} + 10 q^{52} - 2 q^{53} - 4 q^{58} + 12 q^{59} + 2 q^{61} - 28 q^{62} - 16 q^{64} - 7 q^{67} + 12 q^{68} - 16 q^{71} + 5 q^{73} - 14 q^{74} + 28 q^{76} - 4 q^{77} + 11 q^{79} + 8 q^{82} + 8 q^{83} + 18 q^{86} + 6 q^{89} - 25 q^{91} + 16 q^{92} + 12 q^{94} + 4 q^{97} + 4 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^4 + 5 * q^7 - q^11 - 10 * q^13 - 2 * q^14 + 4 * q^16 + 6 * q^17 - 7 * q^19 + 4 * q^22 - 4 * q^23 + 5 * q^25 + 10 * q^26 - 8 * q^28 + 4 * q^29 + 7 * q^31 + 8 * q^32 - 24 * q^34 - 7 * q^37 - 14 * q^38 - 8 * q^41 - 18 * q^43 - 2 * q^44 - 8 * q^46 + 6 * q^47 + 11 * q^49 - 20 * q^50 + 10 * q^52 - 2 * q^53 - 4 * q^58 + 12 * q^59 + 2 * q^61 - 28 * q^62 - 16 * q^64 - 7 * q^67 + 12 * q^68 - 16 * q^71 + 5 * q^73 - 14 * q^74 + 28 * q^76 - 4 * q^77 + 11 * q^79 + 8 * q^82 + 8 * q^83 + 18 * q^86 + 6 * q^89 - 25 * q^91 + 16 * q^92 + 12 * q^94 + 4 * q^97 + 4 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$442$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 + 1.73205i 0 −1.00000 1.73205i 0 0 2.50000 0.866025i 0 0 0
298.1 −1.00000 1.73205i 0 −1.00000 + 1.73205i 0 0 2.50000 + 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.i.a 2
3.b odd 2 1 231.2.i.c 2
7.c even 3 1 inner 693.2.i.a 2
7.c even 3 1 4851.2.a.r 1
7.d odd 6 1 4851.2.a.s 1
21.g even 6 1 1617.2.a.b 1
21.h odd 6 1 231.2.i.c 2
21.h odd 6 1 1617.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.c 2 3.b odd 2 1
231.2.i.c 2 21.h odd 6 1
693.2.i.a 2 1.a even 1 1 trivial
693.2.i.a 2 7.c even 3 1 inner
1617.2.a.a 1 21.h odd 6 1
1617.2.a.b 1 21.g even 6 1
4851.2.a.r 1 7.c even 3 1
4851.2.a.s 1 7.d odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} + 2T_{2} + 4$$ T2^2 + 2*T2 + 4 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} + T + 1$$
$13$ $$(T + 5)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 7T + 49$$
$37$ $$T^{2} + 7T + 49$$
$41$ $$(T + 4)^{2}$$
$43$ $$(T + 9)^{2}$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} + 2T + 4$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 5T + 25$$
$79$ $$T^{2} - 11T + 121$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T - 2)^{2}$$