# Properties

 Label 231.2.a.d Level $231$ Weight $2$ Character orbit 231.a Self dual yes Analytic conductor $1.845$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [231,2,Mod(1,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, 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("231.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 231.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: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.837.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 1$$ x^3 - 6*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} + \beta_1) q^{5} - \beta_1 q^{6} - q^{7} + (2 \beta_1 + 1) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 - q^3 + (b2 + 2) * q^4 + (-b2 + b1) * q^5 - b1 * q^6 - q^7 + (2*b1 + 1) * q^8 + q^9 $$q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} + \beta_1) q^{5} - \beta_1 q^{6} - q^{7} + (2 \beta_1 + 1) q^{8} + q^{9} + (\beta_{2} - 2 \beta_1 + 3) q^{10} + q^{11} + ( - \beta_{2} - 2) q^{12} + ( - \beta_{2} + \beta_1) q^{13} - \beta_1 q^{14} + (\beta_{2} - \beta_1) q^{15} + (\beta_1 + 4) q^{16} - 2 \beta_1 q^{17} + \beta_1 q^{18} + ( - \beta_{2} - \beta_1 + 4) q^{19} + (3 \beta_1 - 7) q^{20} + q^{21} + \beta_1 q^{22} + ( - 2 \beta_1 - 2) q^{23} + ( - 2 \beta_1 - 1) q^{24} + ( - \beta_{2} - 3 \beta_1 + 5) q^{25} + (\beta_{2} - 2 \beta_1 + 3) q^{26} - q^{27} + ( - \beta_{2} - 2) q^{28} + (\beta_{2} - \beta_1 + 4) q^{29} + ( - \beta_{2} + 2 \beta_1 - 3) q^{30} + (2 \beta_{2} - 2) q^{31} + (\beta_{2} + 2) q^{32} - q^{33} + ( - 2 \beta_{2} - 8) q^{34} + (\beta_{2} - \beta_1) q^{35} + (\beta_{2} + 2) q^{36} + (\beta_{2} + 3 \beta_1) q^{37} + ( - \beta_{2} + 2 \beta_1 - 5) q^{38} + (\beta_{2} - \beta_1) q^{39} + (\beta_{2} - 3 \beta_1 + 6) q^{40} + (2 \beta_{2} + 2 \beta_1 + 2) q^{41} + \beta_1 q^{42} + ( - 2 \beta_1 + 2) q^{43} + (\beta_{2} + 2) q^{44} + ( - \beta_{2} + \beta_1) q^{45} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{46} + (\beta_{2} + \beta_1 - 8) q^{47} + ( - \beta_1 - 4) q^{48} + q^{49} + ( - 3 \beta_{2} + 3 \beta_1 - 13) q^{50} + 2 \beta_1 q^{51} + (3 \beta_1 - 7) q^{52} - 2 \beta_{2} q^{53} - \beta_1 q^{54} + ( - \beta_{2} + \beta_1) q^{55} + ( - 2 \beta_1 - 1) q^{56} + (\beta_{2} + \beta_1 - 4) q^{57} + ( - \beta_{2} + 6 \beta_1 - 3) q^{58} + ( - 3 \beta_{2} + \beta_1 - 8) q^{59} + ( - 3 \beta_1 + 7) q^{60} + 6 q^{61} + (2 \beta_1 + 2) q^{62} - q^{63} + (2 \beta_1 - 7) q^{64} + ( - \beta_{2} - 3 \beta_1 + 10) q^{65} - \beta_1 q^{66} + (\beta_{2} + \beta_1 + 4) q^{67} + ( - 8 \beta_1 - 2) q^{68} + (2 \beta_1 + 2) q^{69} + ( - \beta_{2} + 2 \beta_1 - 3) q^{70} + ( - 4 \beta_1 + 4) q^{71} + (2 \beta_1 + 1) q^{72} + (\beta_{2} - \beta_1 + 8) q^{73} + (3 \beta_{2} + 2 \beta_1 + 13) q^{74} + (\beta_{2} + 3 \beta_1 - 5) q^{75} + (4 \beta_{2} - 5 \beta_1 - 1) q^{76} - q^{77} + ( - \beta_{2} + 2 \beta_1 - 3) q^{78} + (4 \beta_1 + 4) q^{79} + ( - 3 \beta_{2} + 2 \beta_1 + 3) q^{80} + q^{81} + (2 \beta_{2} + 6 \beta_1 + 10) q^{82} + (2 \beta_{2} - 6) q^{83} + (\beta_{2} + 2) q^{84} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{85} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{86} + ( - \beta_{2} + \beta_1 - 4) q^{87} + (2 \beta_1 + 1) q^{88} + (4 \beta_{2} + 6) q^{89} + (\beta_{2} - 2 \beta_1 + 3) q^{90} + (\beta_{2} - \beta_1) q^{91} + ( - 2 \beta_{2} - 8 \beta_1 - 6) q^{92} + ( - 2 \beta_{2} + 2) q^{93} + (\beta_{2} - 6 \beta_1 + 5) q^{94} + ( - 7 \beta_{2} + 5 \beta_1 + 4) q^{95} + ( - \beta_{2} - 2) q^{96} + (2 \beta_{2} + 8) q^{97} + \beta_1 q^{98} + q^{99}+O(q^{100})$$ q + b1 * q^2 - q^3 + (b2 + 2) * q^4 + (-b2 + b1) * q^5 - b1 * q^6 - q^7 + (2*b1 + 1) * q^8 + q^9 + (b2 - 2*b1 + 3) * q^10 + q^11 + (-b2 - 2) * q^12 + (-b2 + b1) * q^13 - b1 * q^14 + (b2 - b1) * q^15 + (b1 + 4) * q^16 - 2*b1 * q^17 + b1 * q^18 + (-b2 - b1 + 4) * q^19 + (3*b1 - 7) * q^20 + q^21 + b1 * q^22 + (-2*b1 - 2) * q^23 + (-2*b1 - 1) * q^24 + (-b2 - 3*b1 + 5) * q^25 + (b2 - 2*b1 + 3) * q^26 - q^27 + (-b2 - 2) * q^28 + (b2 - b1 + 4) * q^29 + (-b2 + 2*b1 - 3) * q^30 + (2*b2 - 2) * q^31 + (b2 + 2) * q^32 - q^33 + (-2*b2 - 8) * q^34 + (b2 - b1) * q^35 + (b2 + 2) * q^36 + (b2 + 3*b1) * q^37 + (-b2 + 2*b1 - 5) * q^38 + (b2 - b1) * q^39 + (b2 - 3*b1 + 6) * q^40 + (2*b2 + 2*b1 + 2) * q^41 + b1 * q^42 + (-2*b1 + 2) * q^43 + (b2 + 2) * q^44 + (-b2 + b1) * q^45 + (-2*b2 - 2*b1 - 8) * q^46 + (b2 + b1 - 8) * q^47 + (-b1 - 4) * q^48 + q^49 + (-3*b2 + 3*b1 - 13) * q^50 + 2*b1 * q^51 + (3*b1 - 7) * q^52 - 2*b2 * q^53 - b1 * q^54 + (-b2 + b1) * q^55 + (-2*b1 - 1) * q^56 + (b2 + b1 - 4) * q^57 + (-b2 + 6*b1 - 3) * q^58 + (-3*b2 + b1 - 8) * q^59 + (-3*b1 + 7) * q^60 + 6 * q^61 + (2*b1 + 2) * q^62 - q^63 + (2*b1 - 7) * q^64 + (-b2 - 3*b1 + 10) * q^65 - b1 * q^66 + (b2 + b1 + 4) * q^67 + (-8*b1 - 2) * q^68 + (2*b1 + 2) * q^69 + (-b2 + 2*b1 - 3) * q^70 + (-4*b1 + 4) * q^71 + (2*b1 + 1) * q^72 + (b2 - b1 + 8) * q^73 + (3*b2 + 2*b1 + 13) * q^74 + (b2 + 3*b1 - 5) * q^75 + (4*b2 - 5*b1 - 1) * q^76 - q^77 + (-b2 + 2*b1 - 3) * q^78 + (4*b1 + 4) * q^79 + (-3*b2 + 2*b1 + 3) * q^80 + q^81 + (2*b2 + 6*b1 + 10) * q^82 + (2*b2 - 6) * q^83 + (b2 + 2) * q^84 + (-2*b2 + 4*b1 - 6) * q^85 + (-2*b2 + 2*b1 - 8) * q^86 + (-b2 + b1 - 4) * q^87 + (2*b1 + 1) * q^88 + (4*b2 + 6) * q^89 + (b2 - 2*b1 + 3) * q^90 + (b2 - b1) * q^91 + (-2*b2 - 8*b1 - 6) * q^92 + (-2*b2 + 2) * q^93 + (b2 - 6*b1 + 5) * q^94 + (-7*b2 + 5*b1 + 4) * q^95 + (-b2 - 2) * q^96 + (2*b2 + 8) * q^97 + b1 * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^9 $$3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} + 12 q^{16} + 12 q^{19} - 21 q^{20} + 3 q^{21} - 6 q^{23} - 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} - 6 q^{28} + 12 q^{29} - 9 q^{30} - 6 q^{31} + 6 q^{32} - 3 q^{33} - 24 q^{34} + 6 q^{36} - 15 q^{38} + 18 q^{40} + 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} - 39 q^{50} - 21 q^{52} - 3 q^{56} - 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} + 3 q^{72} + 24 q^{73} + 39 q^{74} - 15 q^{75} - 3 q^{76} - 3 q^{77} - 9 q^{78} + 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} - 18 q^{83} + 6 q^{84} - 18 q^{85} - 24 q^{86} - 12 q^{87} + 3 q^{88} + 18 q^{89} + 9 q^{90} - 18 q^{92} + 6 q^{93} + 15 q^{94} + 12 q^{95} - 6 q^{96} + 24 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^9 + 9 * q^10 + 3 * q^11 - 6 * q^12 + 12 * q^16 + 12 * q^19 - 21 * q^20 + 3 * q^21 - 6 * q^23 - 3 * q^24 + 15 * q^25 + 9 * q^26 - 3 * q^27 - 6 * q^28 + 12 * q^29 - 9 * q^30 - 6 * q^31 + 6 * q^32 - 3 * q^33 - 24 * q^34 + 6 * q^36 - 15 * q^38 + 18 * q^40 + 6 * q^41 + 6 * q^43 + 6 * q^44 - 24 * q^46 - 24 * q^47 - 12 * q^48 + 3 * q^49 - 39 * q^50 - 21 * q^52 - 3 * q^56 - 12 * q^57 - 9 * q^58 - 24 * q^59 + 21 * q^60 + 18 * q^61 + 6 * q^62 - 3 * q^63 - 21 * q^64 + 30 * q^65 + 12 * q^67 - 6 * q^68 + 6 * q^69 - 9 * q^70 + 12 * q^71 + 3 * q^72 + 24 * q^73 + 39 * q^74 - 15 * q^75 - 3 * q^76 - 3 * q^77 - 9 * q^78 + 12 * q^79 + 9 * q^80 + 3 * q^81 + 30 * q^82 - 18 * q^83 + 6 * q^84 - 18 * q^85 - 24 * q^86 - 12 * q^87 + 3 * q^88 + 18 * q^89 + 9 * q^90 - 18 * q^92 + 6 * q^93 + 15 * q^94 + 12 * q^95 - 6 * q^96 + 24 * q^97 + 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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
 −2.36147 −0.167449 2.52892
−2.36147 −1.00000 3.57653 −3.93800 2.36147 −1.00000 −3.72294 1.00000 9.29947
1.2 −0.167449 −1.00000 −1.97196 3.80451 0.167449 −1.00000 0.665102 1.00000 −0.637062
1.3 2.52892 −1.00000 4.39543 0.133492 −2.52892 −1.00000 6.05784 1.00000 0.337590
 $$n$$: 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 231.2.a.d 3
3.b odd 2 1 693.2.a.m 3
4.b odd 2 1 3696.2.a.bp 3
5.b even 2 1 5775.2.a.bw 3
7.b odd 2 1 1617.2.a.s 3
11.b odd 2 1 2541.2.a.bi 3
21.c even 2 1 4851.2.a.bp 3
33.d even 2 1 7623.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.d 3 1.a even 1 1 trivial
693.2.a.m 3 3.b odd 2 1
1617.2.a.s 3 7.b odd 2 1
2541.2.a.bi 3 11.b odd 2 1
3696.2.a.bp 3 4.b odd 2 1
4851.2.a.bp 3 21.c even 2 1
5775.2.a.bw 3 5.b even 2 1
7623.2.a.cb 3 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 6T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(231))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 6T - 1$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3} - 15T + 2$$
$7$ $$(T + 1)^{3}$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} - 15T + 2$$
$17$ $$T^{3} - 24T + 8$$
$19$ $$T^{3} - 12 T^{2} + \cdots + 36$$
$23$ $$T^{3} + 6 T^{2} + \cdots - 32$$
$29$ $$T^{3} - 12 T^{2} + \cdots - 6$$
$31$ $$T^{3} + 6 T^{2} + \cdots + 32$$
$37$ $$T^{3} - 75T - 246$$
$41$ $$T^{3} - 6 T^{2} + \cdots + 32$$
$43$ $$T^{3} - 6 T^{2} + \cdots + 48$$
$47$ $$T^{3} + 24 T^{2} + \cdots + 328$$
$53$ $$T^{3} - 48T - 120$$
$59$ $$T^{3} + 24 T^{2} + \cdots - 716$$
$61$ $$(T - 6)^{3}$$
$67$ $$T^{3} - 12 T^{2} + \cdots + 4$$
$71$ $$T^{3} - 12 T^{2} + \cdots + 384$$
$73$ $$T^{3} - 24 T^{2} + \cdots - 394$$
$79$ $$T^{3} - 12 T^{2} + \cdots + 256$$
$83$ $$T^{3} + 18 T^{2} + \cdots + 48$$
$89$ $$T^{3} - 18 T^{2} + \cdots + 1896$$
$97$ $$T^{3} - 24 T^{2} + \cdots - 8$$