# Properties

 Label 1170.2.e.f 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: 6.0.3534400.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40$$ x^6 - 2*x^5 - 3*x^4 + 16*x^3 + x^2 - 12*x + 40 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 130) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} - q^{4} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{5}+ \cdots - \beta_{5} q^{8}+O(q^{10})$$ q + b5 * q^2 - q^4 + (b5 - b4 + b3 - b2 - b1) * q^5 + (-b4 + b2 + b1) * q^7 - b5 * q^8 $$q + \beta_{5} q^{2} - q^{4} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{5}+ \cdots + ( - 5 \beta_{5} + 3 \beta_{4} + \cdots + \beta_1) q^{98}+O(q^{100})$$ q + b5 * q^2 - q^4 + (b5 - b4 + b3 - b2 - b1) * q^5 + (-b4 + b2 + b1) * q^7 - b5 * q^8 + (-b3 - b2 - 1) * q^10 - 2 * q^11 + b5 * q^13 + (-b4 + b3 - 2*b1) * q^14 + q^16 + (-b4 - b3 + 2*b2 + 2*b1) * q^17 + (-2*b3 - 2*b2 + 2*b1 - 2) * q^19 + (-b5 + b4 - b3 + b2 + b1) * q^20 - 2*b5 * q^22 + (-2*b5 + 2*b4 - 2*b3) * q^23 + (-2*b5 + b4 - b2 - 3) * q^25 - q^26 + (b4 - b2 - b1) * q^28 + (-2*b3 - 2*b2 + 2*b1 - 2) * q^29 + (-2*b4 - 2*b2 - 2*b1 + 4) * q^31 + b5 * q^32 + (-b4 + 2*b3 + b2 - 3*b1) * q^34 + (-2*b5 - b4 - b2 - 4*b1 + 2) * q^35 + (2*b5 + b4 - 2*b3 + b2 + b1) * q^37 + (-2*b5 - 2*b3 + 2*b2 + 2*b1) * q^38 + (b3 + b2 + 1) * q^40 + (-2*b4 - 2*b3 - 4*b2 - 2) * q^41 + (-4*b5 - 3*b4 + 3*b3) * q^43 + 2 * q^44 + (2*b4 + 2*b2 + 2*b1 + 2) * q^46 + (-4*b5 - b4 + b2 + b1) * q^47 + (-3*b4 - b3 - 4*b2 - 2*b1 - 5) * q^49 + (-3*b5 - b3 + 2*b1 + 2) * q^50 - b5 * q^52 + (-2*b5 + 2*b4 - 2*b3) * q^53 + (-2*b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1) * q^55 + (b4 - b3 + 2*b1) * q^56 + (-2*b5 - 2*b3 + 2*b2 + 2*b1) * q^58 + (2*b3 + 2*b2 - 2*b1 + 2) * q^59 + (-2*b4 - 2*b3 - 4*b2 + 2) * q^61 + (4*b5 + 2*b4 - 2*b3) * q^62 - q^64 + (-b3 - b2 - 1) * q^65 + (4*b5 + 2*b4 - 2*b3) * q^67 + (b4 + b3 - 2*b2 - 2*b1) * q^68 + (2*b5 + 4*b4 - b3 + 2) * q^70 + (b4 + 2*b3 + 3*b2 - b1 + 4) * q^71 - 6*b5 * q^73 + (b4 + b3 + 2*b2 - 2) * q^74 + (2*b3 + 2*b2 - 2*b1 + 2) * q^76 + (2*b4 - 2*b2 - 2*b1) * q^77 + (2*b3 + 2*b2 - 2*b1 - 8) * q^79 + (b5 - b4 + b3 - b2 - b1) * q^80 + (-2*b5 + 2*b4 - 4*b3 + 2*b2 + 2*b1) * q^82 + (4*b5 - 2*b3 + 2*b2 + 2*b1) * q^83 + (-2*b5 - 2*b4 - b3 - 2*b2 - 6*b1 + 6) * q^85 + (-3*b4 - 3*b2 - 3*b1 + 4) * q^86 + 2*b5 * q^88 + (-2*b3 - 2*b2 + 2*b1 - 2) * q^89 + (-b4 + b3 - 2*b1) * q^91 + (2*b5 - 2*b4 + 2*b3) * q^92 + (-b4 + b3 - 2*b1 + 4) * q^94 + (-10*b5 - 4*b4 + 2*b1) * q^95 + (10*b5 - 4*b4 + 6*b3 - 2*b2 - 2*b1) * q^97 + (-5*b5 + 3*b4 - 4*b3 + b2 + b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4}+O(q^{10})$$ 6 * q - 6 * q^4 $$6 q - 6 q^{4} - 4 q^{10} - 12 q^{11} - 4 q^{14} + 6 q^{16} - 4 q^{19} - 16 q^{25} - 6 q^{26} - 4 q^{29} + 24 q^{31} - 8 q^{34} + 6 q^{35} + 4 q^{40} - 4 q^{41} + 12 q^{44} + 12 q^{46} - 26 q^{49} + 16 q^{50} + 4 q^{56} + 4 q^{59} + 20 q^{61} - 6 q^{64} - 4 q^{65} + 12 q^{70} + 16 q^{71} - 16 q^{74} + 4 q^{76} - 56 q^{79} + 28 q^{85} + 24 q^{86} - 4 q^{89} - 4 q^{91} + 20 q^{94} + 4 q^{95}+O(q^{100})$$ 6 * q - 6 * q^4 - 4 * q^10 - 12 * q^11 - 4 * q^14 + 6 * q^16 - 4 * q^19 - 16 * q^25 - 6 * q^26 - 4 * q^29 + 24 * q^31 - 8 * q^34 + 6 * q^35 + 4 * q^40 - 4 * q^41 + 12 * q^44 + 12 * q^46 - 26 * q^49 + 16 * q^50 + 4 * q^56 + 4 * q^59 + 20 * q^61 - 6 * q^64 - 4 * q^65 + 12 * q^70 + 16 * q^71 - 16 * q^74 + 4 * q^76 - 56 * q^79 + 28 * q^85 + 24 * q^86 - 4 * q^89 - 4 * q^91 + 20 * q^94 + 4 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 4\nu^{5} - 70\nu^{4} + 183\nu^{3} + 120\nu^{2} - 966\nu + 240 ) / 445$$ (4*v^5 - 70*v^4 + 183*v^3 + 120*v^2 - 966*v + 240) / 445 $$\beta_{3}$$ $$=$$ $$( 13\nu^{5} - 5\nu^{4} - 184\nu^{3} + 390\nu^{2} + 643\nu - 1000 ) / 445$$ (13*v^5 - 5*v^4 - 184*v^3 + 390*v^2 + 643*v - 1000) / 445 $$\beta_{4}$$ $$=$$ $$( -3\nu^{5} + 8\nu^{4} - 26\nu^{3} - \nu^{2} + 57\nu - 180 ) / 89$$ (-3*v^5 + 8*v^4 - 26*v^3 - v^2 + 57*v - 180) / 89 $$\beta_{5}$$ $$=$$ $$( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 178$$ (-9*v^5 + 24*v^4 + 11*v^3 - 92*v^2 - 7*v - 6) / 178
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2$$ 2*b5 - b4 + 2*b3 + b2 + 2 $$\nu^{3}$$ $$=$$ $$4\beta_{5} - 4\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 4$$ 4*b5 - 4*b4 + 2*b3 + b2 + 2*b1 - 4 $$\nu^{4}$$ $$=$$ $$14\beta_{5} - 14\beta_{4} + 9\beta_{3} - 3\beta_{2} - 10\beta _1 - 6$$ 14*b5 - 14*b4 + 9*b3 - 3*b2 - 10*b1 - 6 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - 32\beta_{4} + 6\beta_{3} - 17\beta_{2} - 25\beta _1 - 42$$ 2*b5 - 32*b4 + 6*b3 - 17*b2 - 25*b1 - 42

## 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
 2.19082 − 1.44755i −1.81837 + 0.301352i 0.627553 + 1.14620i 2.19082 + 1.44755i −1.81837 − 0.301352i 0.627553 − 1.14620i
1.00000i 0 −1.00000 −1.44755 1.70429i 0 4.38164i 1.00000i 0 −1.70429 + 1.44755i
469.2 1.00000i 0 −1.00000 0.301352 2.21567i 0 3.63675i 1.00000i 0 −2.21567 0.301352i
469.3 1.00000i 0 −1.00000 1.14620 + 1.91995i 0 1.25511i 1.00000i 0 1.91995 1.14620i
469.4 1.00000i 0 −1.00000 −1.44755 + 1.70429i 0 4.38164i 1.00000i 0 −1.70429 1.44755i
469.5 1.00000i 0 −1.00000 0.301352 + 2.21567i 0 3.63675i 1.00000i 0 −2.21567 + 0.301352i
469.6 1.00000i 0 −1.00000 1.14620 1.91995i 0 1.25511i 1.00000i 0 1.91995 + 1.14620i
 $$n$$: 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.f 6
3.b odd 2 1 130.2.b.a 6
5.b even 2 1 inner 1170.2.e.f 6
5.c odd 4 1 5850.2.a.cp 3
5.c odd 4 1 5850.2.a.cs 3
12.b even 2 1 1040.2.d.b 6
15.d odd 2 1 130.2.b.a 6
15.e even 4 1 650.2.a.n 3
15.e even 4 1 650.2.a.o 3
39.d odd 2 1 1690.2.b.a 6
39.f even 4 1 1690.2.c.a 6
39.f even 4 1 1690.2.c.d 6
60.h even 2 1 1040.2.d.b 6
60.l odd 4 1 5200.2.a.ce 3
60.l odd 4 1 5200.2.a.cf 3
195.e odd 2 1 1690.2.b.a 6
195.n even 4 1 1690.2.c.a 6
195.n even 4 1 1690.2.c.d 6
195.s even 4 1 8450.2.a.bs 3
195.s even 4 1 8450.2.a.cc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.b.a 6 3.b odd 2 1
130.2.b.a 6 15.d odd 2 1
650.2.a.n 3 15.e even 4 1
650.2.a.o 3 15.e even 4 1
1040.2.d.b 6 12.b even 2 1
1040.2.d.b 6 60.h even 2 1
1170.2.e.f 6 1.a even 1 1 trivial
1170.2.e.f 6 5.b even 2 1 inner
1690.2.b.a 6 39.d odd 2 1
1690.2.b.a 6 195.e odd 2 1
1690.2.c.a 6 39.f even 4 1
1690.2.c.a 6 195.n even 4 1
1690.2.c.d 6 39.f even 4 1
1690.2.c.d 6 195.n even 4 1
5200.2.a.ce 3 60.l odd 4 1
5200.2.a.cf 3 60.l odd 4 1
5850.2.a.cp 3 5.c odd 4 1
5850.2.a.cs 3 5.c odd 4 1
8450.2.a.bs 3 195.s even 4 1
8450.2.a.cc 3 195.s even 4 1

## Hecke kernels

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

 $$T_{7}^{6} + 34T_{7}^{4} + 305T_{7}^{2} + 400$$ T7^6 + 34*T7^4 + 305*T7^2 + 400 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 8 T^{4} + \cdots + 125$$
$7$ $$T^{6} + 34 T^{4} + \cdots + 400$$
$11$ $$(T + 2)^{6}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 102 T^{4} + \cdots + 35344$$
$19$ $$(T^{3} + 2 T^{2} - 44 T + 40)^{2}$$
$23$ $$T^{6} + 68 T^{4} + \cdots + 256$$
$29$ $$(T^{3} + 2 T^{2} - 44 T + 40)^{2}$$
$31$ $$(T^{3} - 12 T^{2} + \cdots + 80)^{2}$$
$37$ $$T^{6} + 62 T^{4} + \cdots + 4$$
$41$ $$(T^{3} + 2 T^{2} + \cdots - 320)^{2}$$
$43$ $$T^{6} + 174 T^{4} + \cdots + 87616$$
$47$ $$T^{6} + 66 T^{4} + \cdots + 64$$
$53$ $$T^{6} + 68 T^{4} + \cdots + 256$$
$59$ $$(T^{3} - 2 T^{2} - 44 T - 40)^{2}$$
$61$ $$(T^{3} - 10 T^{2} + \cdots - 32)^{2}$$
$67$ $$T^{6} + 104 T^{4} + \cdots + 6400$$
$71$ $$(T^{3} - 8 T^{2} + \cdots + 200)^{2}$$
$73$ $$(T^{2} + 36)^{3}$$
$79$ $$(T^{3} + 28 T^{2} + \cdots + 320)^{2}$$
$83$ $$T^{6} + 176 T^{4} + \cdots + 25600$$
$89$ $$(T^{3} + 2 T^{2} - 44 T + 40)^{2}$$
$97$ $$T^{6} + 572 T^{4} + \cdots + 2534464$$