# Properties

 Label 231.2.i.c Level $231$ Weight $2$ Character orbit 231.i Analytic conductor $1.845$ 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] = [231,2,Mod(67,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("231.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 231.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: $$1.84454428669$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + 2 q^{6} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + 2*z * q^2 + (-z + 1) * q^3 + (2*z - 2) * q^4 + 2 * q^6 + (z + 2) * q^7 - z * q^9 $$q + 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + 2 q^{6} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} + 2 \zeta_{6} q^{12} - 5 q^{13} + (6 \zeta_{6} - 2) q^{14} + 4 \zeta_{6} q^{16} + (6 \zeta_{6} - 6) q^{17} + ( - 2 \zeta_{6} + 2) q^{18} - 7 \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 3) q^{21} + 2 q^{22} + 4 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - 10 \zeta_{6} q^{26} - q^{27} + (4 \zeta_{6} - 6) q^{28} - 2 q^{29} + ( - 7 \zeta_{6} + 7) q^{31} + (8 \zeta_{6} - 8) q^{32} - \zeta_{6} q^{33} - 12 q^{34} + 2 q^{36} - 7 \zeta_{6} q^{37} + ( - 14 \zeta_{6} + 14) q^{38} + (5 \zeta_{6} - 5) q^{39} + 4 q^{41} + (2 \zeta_{6} + 4) q^{42} - 9 q^{43} + 2 \zeta_{6} q^{44} + (8 \zeta_{6} - 8) q^{46} - 6 \zeta_{6} q^{47} + 4 q^{48} + (5 \zeta_{6} + 3) q^{49} + 10 q^{50} + 6 \zeta_{6} q^{51} + ( - 10 \zeta_{6} + 10) q^{52} + ( - 2 \zeta_{6} + 2) q^{53} - 2 \zeta_{6} q^{54} - 7 q^{57} - 4 \zeta_{6} q^{58} + (12 \zeta_{6} - 12) q^{59} + 2 \zeta_{6} q^{61} + 14 q^{62} + ( - 3 \zeta_{6} + 1) q^{63} - 8 q^{64} + ( - 2 \zeta_{6} + 2) q^{66} + (7 \zeta_{6} - 7) q^{67} - 12 \zeta_{6} q^{68} + 4 q^{69} + 8 q^{71} + ( - 5 \zeta_{6} + 5) q^{73} + ( - 14 \zeta_{6} + 14) q^{74} - 5 \zeta_{6} q^{75} + 14 q^{76} + ( - 2 \zeta_{6} + 3) q^{77} - 10 q^{78} + 11 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 8 \zeta_{6} q^{82} - 4 q^{83} + (6 \zeta_{6} - 2) q^{84} - 18 \zeta_{6} q^{86} + (2 \zeta_{6} - 2) q^{87} - 6 \zeta_{6} q^{89} + ( - 5 \zeta_{6} - 10) q^{91} - 8 q^{92} - 7 \zeta_{6} q^{93} + ( - 12 \zeta_{6} + 12) q^{94} + 8 \zeta_{6} q^{96} + 2 q^{97} + (16 \zeta_{6} - 10) q^{98} - q^{99} +O(q^{100})$$ q + 2*z * q^2 + (-z + 1) * q^3 + (2*z - 2) * q^4 + 2 * q^6 + (z + 2) * q^7 - z * q^9 + (-z + 1) * q^11 + 2*z * q^12 - 5 * q^13 + (6*z - 2) * q^14 + 4*z * q^16 + (6*z - 6) * q^17 + (-2*z + 2) * q^18 - 7*z * q^19 + (-2*z + 3) * q^21 + 2 * q^22 + 4*z * q^23 + (-5*z + 5) * q^25 - 10*z * q^26 - q^27 + (4*z - 6) * q^28 - 2 * q^29 + (-7*z + 7) * q^31 + (8*z - 8) * q^32 - z * q^33 - 12 * q^34 + 2 * q^36 - 7*z * q^37 + (-14*z + 14) * q^38 + (5*z - 5) * q^39 + 4 * q^41 + (2*z + 4) * q^42 - 9 * q^43 + 2*z * q^44 + (8*z - 8) * q^46 - 6*z * q^47 + 4 * q^48 + (5*z + 3) * q^49 + 10 * q^50 + 6*z * q^51 + (-10*z + 10) * q^52 + (-2*z + 2) * q^53 - 2*z * q^54 - 7 * q^57 - 4*z * q^58 + (12*z - 12) * q^59 + 2*z * q^61 + 14 * q^62 + (-3*z + 1) * q^63 - 8 * q^64 + (-2*z + 2) * q^66 + (7*z - 7) * q^67 - 12*z * q^68 + 4 * q^69 + 8 * q^71 + (-5*z + 5) * q^73 + (-14*z + 14) * q^74 - 5*z * q^75 + 14 * q^76 + (-2*z + 3) * q^77 - 10 * q^78 + 11*z * q^79 + (z - 1) * q^81 + 8*z * q^82 - 4 * q^83 + (6*z - 2) * q^84 - 18*z * q^86 + (2*z - 2) * q^87 - 6*z * q^89 + (-5*z - 10) * q^91 - 8 * q^92 - 7*z * q^93 + (-12*z + 12) * q^94 + 8*z * q^96 + 2 * q^97 + (16*z - 10) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} - 2 q^{4} + 4 q^{6} + 5 q^{7} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 - 2 * q^4 + 4 * q^6 + 5 * q^7 - q^9 $$2 q + 2 q^{2} + q^{3} - 2 q^{4} + 4 q^{6} + 5 q^{7} - q^{9} + q^{11} + 2 q^{12} - 10 q^{13} + 2 q^{14} + 4 q^{16} - 6 q^{17} + 2 q^{18} - 7 q^{19} + 4 q^{21} + 4 q^{22} + 4 q^{23} + 5 q^{25} - 10 q^{26} - 2 q^{27} - 8 q^{28} - 4 q^{29} + 7 q^{31} - 8 q^{32} - q^{33} - 24 q^{34} + 4 q^{36} - 7 q^{37} + 14 q^{38} - 5 q^{39} + 8 q^{41} + 10 q^{42} - 18 q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{47} + 8 q^{48} + 11 q^{49} + 20 q^{50} + 6 q^{51} + 10 q^{52} + 2 q^{53} - 2 q^{54} - 14 q^{57} - 4 q^{58} - 12 q^{59} + 2 q^{61} + 28 q^{62} - q^{63} - 16 q^{64} + 2 q^{66} - 7 q^{67} - 12 q^{68} + 8 q^{69} + 16 q^{71} + 5 q^{73} + 14 q^{74} - 5 q^{75} + 28 q^{76} + 4 q^{77} - 20 q^{78} + 11 q^{79} - q^{81} + 8 q^{82} - 8 q^{83} + 2 q^{84} - 18 q^{86} - 2 q^{87} - 6 q^{89} - 25 q^{91} - 16 q^{92} - 7 q^{93} + 12 q^{94} + 8 q^{96} + 4 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 - 2 * q^4 + 4 * q^6 + 5 * q^7 - q^9 + q^11 + 2 * q^12 - 10 * q^13 + 2 * q^14 + 4 * q^16 - 6 * q^17 + 2 * q^18 - 7 * q^19 + 4 * q^21 + 4 * q^22 + 4 * q^23 + 5 * q^25 - 10 * q^26 - 2 * q^27 - 8 * q^28 - 4 * q^29 + 7 * q^31 - 8 * q^32 - q^33 - 24 * q^34 + 4 * q^36 - 7 * q^37 + 14 * q^38 - 5 * q^39 + 8 * q^41 + 10 * q^42 - 18 * q^43 + 2 * q^44 - 8 * q^46 - 6 * q^47 + 8 * q^48 + 11 * q^49 + 20 * q^50 + 6 * q^51 + 10 * q^52 + 2 * q^53 - 2 * q^54 - 14 * q^57 - 4 * q^58 - 12 * q^59 + 2 * q^61 + 28 * q^62 - q^63 - 16 * q^64 + 2 * q^66 - 7 * q^67 - 12 * q^68 + 8 * q^69 + 16 * q^71 + 5 * q^73 + 14 * q^74 - 5 * q^75 + 28 * q^76 + 4 * q^77 - 20 * q^78 + 11 * q^79 - q^81 + 8 * q^82 - 8 * q^83 + 2 * q^84 - 18 * q^86 - 2 * q^87 - 6 * q^89 - 25 * q^91 - 16 * q^92 - 7 * q^93 + 12 * q^94 + 8 * q^96 + 4 * q^97 - 4 * q^98 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/231\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 + 1.73205i 0.500000 0.866025i −1.00000 + 1.73205i 0 2.00000 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
100.1 1.00000 1.73205i 0.500000 + 0.866025i −1.00000 1.73205i 0 2.00000 2.50000 0.866025i 0 −0.500000 + 0.866025i 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 231.2.i.c 2
3.b odd 2 1 693.2.i.a 2
7.c even 3 1 inner 231.2.i.c 2
7.c even 3 1 1617.2.a.a 1
7.d odd 6 1 1617.2.a.b 1
21.g even 6 1 4851.2.a.s 1
21.h odd 6 1 693.2.i.a 2
21.h odd 6 1 4851.2.a.r 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke characteristic polynomials

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