# Properties

 Label 418.2.a.g Level $418$ Weight $2$ Character orbit 418.a Self dual yes Analytic conductor $3.338$ Analytic rank $1$ 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] = [418,2,Mod(1,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("418.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.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: $$3.33774680449$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 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
 −2.14510 −0.523976 2.66908
−1.00000 −2.14510 1.00000 −1.60147 2.14510 2.89167 −1.00000 1.60147 1.60147
1.2 −1.00000 −0.523976 1.00000 2.72545 0.523976 −4.67750 −1.00000 −2.72545 −2.72545
1.3 −1.00000 2.66908 1.00000 −4.12398 −2.66908 −4.21417 −1.00000 4.12398 4.12398
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$11$$ $$1$$
$$19$$ $$1$$

## 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 418.2.a.g 3
3.b odd 2 1 3762.2.a.bg 3
4.b odd 2 1 3344.2.a.q 3
11.b odd 2 1 4598.2.a.bo 3
19.b odd 2 1 7942.2.a.bi 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.g 3 1.a even 1 1 trivial
3344.2.a.q 3 4.b odd 2 1
3762.2.a.bg 3 3.b odd 2 1
4598.2.a.bo 3 11.b odd 2 1
7942.2.a.bi 3 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 6T_{3} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(418))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} - 6T - 3$$
$5$ $$T^{3} + 3 T^{2} + \cdots - 18$$
$7$ $$T^{3} + 6 T^{2} + \cdots - 57$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3} - 18T - 29$$
$17$ $$T^{3} + 9 T^{2} + \cdots - 4$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} + 15 T^{2} + \cdots + 76$$
$29$ $$T^{3} - 6 T^{2} + \cdots + 147$$
$31$ $$T^{3} + 3 T^{2} + \cdots - 134$$
$37$ $$T^{3} + 6 T^{2} + \cdots - 96$$
$41$ $$T^{3} + 9 T^{2} + \cdots - 122$$
$43$ $$T^{3} + 21 T^{2} + \cdots - 184$$
$47$ $$T^{3} + 12 T^{2} + \cdots - 672$$
$53$ $$T^{3} + 9 T^{2} + \cdots - 452$$
$59$ $$T^{3} + 3 T^{2} + \cdots - 64$$
$61$ $$T^{3} - 24 T^{2} + \cdots - 192$$
$67$ $$T^{3} - 96T + 123$$
$71$ $$T^{3} - 21 T^{2} + \cdots + 1306$$
$73$ $$T^{3} + 3 T^{2} + \cdots - 12$$
$79$ $$T^{3} + 6 T^{2} + \cdots - 944$$
$83$ $$T^{3} + 33 T^{2} + \cdots + 1104$$
$89$ $$T^{3} - 6 T^{2} + \cdots + 144$$
$97$ $$T^{3} - 12 T^{2} + \cdots + 256$$