# Properties

 Label 1170.2.e.g Level $1170$ Weight $2$ Character orbit 1170.e Analytic conductor $9.342$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(469,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1170.469");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.e (of order $$2$$, degree $$1$$, 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: $$9.34249703649$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} - q^{4} + (\beta_{5} - \beta_{2}) q^{5} + (\beta_{4} - \beta_1) q^{7} + \beta_{4} q^{8}+O(q^{10})$$ q - b4 * q^2 - q^4 + (b5 - b2) * q^5 + (b4 - b1) * q^7 + b4 * q^8 $$q - \beta_{4} q^{2} - q^{4} + (\beta_{5} - \beta_{2}) q^{5} + (\beta_{4} - \beta_1) q^{7} + \beta_{4} q^{8} + ( - \beta_{3} + \beta_1) q^{10} - 2 q^{11} - \beta_{4} q^{13} + ( - \beta_{2} + 1) q^{14} + q^{16} + (\beta_{5} - \beta_{4} + \cdots + 3 \beta_1) q^{17}+ \cdots + (\beta_{5} - 3 \beta_{4} + \cdots - 2 \beta_1) q^{98}+O(q^{100})$$ q - b4 * q^2 - q^4 + (b5 - b2) * q^5 + (b4 - b1) * q^7 + b4 * q^8 + (-b3 + b1) * q^10 - 2 * q^11 - b4 * q^13 + (-b2 + 1) * q^14 + q^16 + (b5 - b4 - b3 + 3*b1) * q^17 + (-b5 - b3 + b2 - 1) * q^19 + (-b5 + b2) * q^20 + 2*b4 * q^22 + (2*b5 - b4 - 2*b3 + b1) * q^23 + (b5 - 2*b4 - b3 + 2*b2 + 1) * q^25 - q^26 + (-b4 + b1) * q^28 + (2*b5 + 2*b3 + b2 + 5) * q^29 + (-b5 - b3 + 2*b2 - 2) * q^31 - b4 * q^32 + (-b5 - b3 + 3*b2 - 1) * q^34 + (b5 + 2*b4 + b3 + b2 - 1) * q^35 + (2*b5 + 2*b4 - 2*b3) * q^37 + (-b5 + b4 + b3 - b1) * q^38 + (b3 - b1) * q^40 + (b5 + b3 - 4) * q^41 + (3*b5 - 2*b4 - 3*b3) * q^43 + 2 * q^44 + (-2*b5 - 2*b3 + b2 - 1) * q^46 + (2*b4 + 2*b1) * q^47 + (b5 + b3 + 2*b2 + 3) * q^49 + (-b5 - b4 - b3 - 2*b1 - 2) * q^50 + b4 * q^52 + (-b5 + 2*b4 + b3 + 4*b1) * q^53 + (-2*b5 + 2*b2) * q^55 + (b2 - 1) * q^56 + (2*b5 - 5*b4 - 2*b3 - b1) * q^58 + (-2*b5 - 2*b3 + 2*b2 + 4) * q^59 + (-2*b5 - 2*b3 + 6*b2) * q^61 + (-b5 + 2*b4 + b3 - 2*b1) * q^62 - q^64 + (-b3 + b1) * q^65 + (2*b5 + 2*b4 - 2*b3 + 4*b1) * q^67 + (-b5 + b4 + b3 - 3*b1) * q^68 + (b5 + b4 - b3 - b1 + 2) * q^70 + (3*b5 + 3*b3 - 2*b2) * q^71 + (3*b4 - 3*b1) * q^73 + (-2*b5 - 2*b3 + 2) * q^74 + (b5 + b3 - b2 + 1) * q^76 + (-2*b4 + 2*b1) * q^77 + (-5*b5 - 5*b3 + 2*b2) * q^79 + (b5 - b2) * q^80 + (b5 + 4*b4 - b3) * q^82 + (-b5 - 2*b4 + b3 - 4*b1) * q^83 + (-2*b5 - 8*b4 - 2*b3 - b2 - 2*b1 - 1) * q^85 + (-3*b5 - 3*b3 - 2) * q^86 - 2*b4 * q^88 + (2*b5 + 2*b3 - 2*b2 + 8) * q^89 + (-b2 + 1) * q^91 + (-2*b5 + b4 + 2*b3 - b1) * q^92 + (2*b2 + 2) * q^94 + (-2*b5 + 3*b4 - b1 - 4) * q^95 + (9*b4 - b1) * q^97 + (b5 - 3*b4 - b3 - 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4}+O(q^{10})$$ 6 * q - 6 * q^4 $$6 q - 6 q^{4} + 2 q^{10} - 12 q^{11} + 8 q^{14} + 6 q^{16} - 4 q^{19} + 2 q^{25} - 6 q^{26} + 20 q^{29} - 12 q^{31} - 8 q^{34} - 12 q^{35} - 2 q^{40} - 28 q^{41} + 12 q^{44} + 10 q^{49} - 8 q^{50} - 8 q^{56} + 28 q^{59} - 4 q^{61} - 6 q^{64} + 2 q^{65} + 12 q^{70} - 8 q^{71} + 20 q^{74} + 4 q^{76} + 16 q^{79} + 4 q^{85} + 44 q^{89} + 8 q^{91} + 8 q^{94} - 20 q^{95}+O(q^{100})$$ 6 * q - 6 * q^4 + 2 * q^10 - 12 * q^11 + 8 * q^14 + 6 * q^16 - 4 * q^19 + 2 * q^25 - 6 * q^26 + 20 * q^29 - 12 * q^31 - 8 * q^34 - 12 * q^35 - 2 * q^40 - 28 * q^41 + 12 * q^44 + 10 * q^49 - 8 * q^50 - 8 * q^56 + 28 * q^59 - 4 * q^61 - 6 * q^64 + 2 * q^65 + 12 * q^70 - 8 * q^71 + 20 * q^74 + 4 * q^76 + 16 * q^79 + 4 * q^85 + 44 * q^89 + 8 * q^91 + 8 * q^94 - 20 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23$$ (-3*v^5 + v^4 + 11*v^3 - 26*v^2 + 6*v - 1) / 23 $$\beta_{2}$$ $$=$$ $$( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23$$ (-4*v^5 + 9*v^4 - 16*v^3 - 4*v^2 + 8*v - 9) / 23 $$\beta_{3}$$ $$=$$ $$( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23$$ (6*v^5 - 2*v^4 + v^3 + 6*v^2 + 80*v + 2) / 23 $$\beta_{4}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{5}$$ $$=$$ $$( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23$$ (-16*v^5 + 36*v^4 - 41*v^3 - 16*v^2 - 60*v + 56) / 23
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2$$ (b4 + b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2$$ (b5 + 4*b4 - b3 + 2*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2$$ b5 + 2*b4 - 2*b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7$$ 2*b5 + 2*b3 - 5*b2 - 7 $$\nu^{5}$$ $$=$$ $$-9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9$$ -9*b4 + 5*b3 - 8*b2 - 8*b1 - 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$$.

 $$n$$ $$911$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
469.1
 1.45161 + 1.45161i −0.854638 − 0.854638i 0.403032 + 0.403032i 1.45161 − 1.45161i −0.854638 + 0.854638i 0.403032 − 0.403032i
1.00000i 0 −1.00000 −2.21432 + 0.311108i 0 0.903212i 1.00000i 0 0.311108 + 2.21432i
469.2 1.00000i 0 −1.00000 0.539189 + 2.17009i 0 3.70928i 1.00000i 0 2.17009 0.539189i
469.3 1.00000i 0 −1.00000 1.67513 1.48119i 0 1.19394i 1.00000i 0 −1.48119 1.67513i
469.4 1.00000i 0 −1.00000 −2.21432 0.311108i 0 0.903212i 1.00000i 0 0.311108 2.21432i
469.5 1.00000i 0 −1.00000 0.539189 2.17009i 0 3.70928i 1.00000i 0 2.17009 + 0.539189i
469.6 1.00000i 0 −1.00000 1.67513 + 1.48119i 0 1.19394i 1.00000i 0 −1.48119 + 1.67513i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 469.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.e.g 6
3.b odd 2 1 1170.2.e.h yes 6
5.b even 2 1 inner 1170.2.e.g 6
5.c odd 4 1 5850.2.a.cq 3
5.c odd 4 1 5850.2.a.cr 3
15.d odd 2 1 1170.2.e.h yes 6
15.e even 4 1 5850.2.a.co 3
15.e even 4 1 5850.2.a.ct 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.e.g 6 1.a even 1 1 trivial
1170.2.e.g 6 5.b even 2 1 inner
1170.2.e.h yes 6 3.b odd 2 1
1170.2.e.h yes 6 15.d odd 2 1
5850.2.a.co 3 15.e even 4 1
5850.2.a.cq 3 5.c odd 4 1
5850.2.a.cr 3 5.c odd 4 1
5850.2.a.ct 3 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1170, [\chi])$$:

 $$T_{7}^{6} + 16T_{7}^{4} + 32T_{7}^{2} + 16$$ T7^6 + 16*T7^4 + 32*T7^2 + 16 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{4} + \cdots + 125$$
$7$ $$T^{6} + 16 T^{4} + \cdots + 16$$
$11$ $$(T + 2)^{6}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 72 T^{4} + \cdots + 13456$$
$19$ $$(T^{3} + 2 T^{2} - 8 T + 4)^{2}$$
$23$ $$T^{6} + 80 T^{4} + \cdots + 5776$$
$29$ $$(T^{3} - 10 T^{2} + \cdots + 388)^{2}$$
$31$ $$(T^{3} + 6 T^{2} - 4 T - 40)^{2}$$
$37$ $$T^{6} + 140 T^{4} + \cdots + 18496$$
$41$ $$(T^{3} + 14 T^{2} + \cdots + 40)^{2}$$
$43$ $$T^{6} + 240 T^{4} + \cdots + 256$$
$47$ $$T^{6} + 48 T^{4} + \cdots + 1024$$
$53$ $$T^{6} + 272 T^{4} + \cdots + 246016$$
$59$ $$(T^{3} - 14 T^{2} + \cdots + 152)^{2}$$
$61$ $$(T^{3} + 2 T^{2} + \cdots - 680)^{2}$$
$67$ $$T^{6} + 140 T^{4} + \cdots + 64$$
$71$ $$(T^{3} + 4 T^{2} + \cdots - 400)^{2}$$
$73$ $$T^{6} + 144 T^{4} + \cdots + 11664$$
$79$ $$(T^{3} - 8 T^{2} + \cdots + 1712)^{2}$$
$83$ $$T^{6} + 128 T^{4} + \cdots + 6400$$
$89$ $$(T^{3} - 22 T^{2} + \cdots - 200)^{2}$$
$97$ $$T^{6} + 272 T^{4} + \cdots + 583696$$