# Properties

 Label 570.2.i.j Level $570$ Weight $2$ Character orbit 570.i Analytic conductor $4.551$ 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] = [570,2,Mod(121,570)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(570, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("570.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.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: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.29654208.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 14x^{4} + 49x^{2} + 12$$ x^6 + 14*x^4 + 49*x^2 + 12 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_1 - 1) q^{3} - \beta_1 q^{4} + (\beta_1 - 1) q^{5} - \beta_1 q^{6} - \beta_{3} q^{7} + q^{8} - \beta_1 q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b1 - 1) * q^3 - b1 * q^4 + (b1 - 1) * q^5 - b1 * q^6 - b3 * q^7 + q^8 - b1 * q^9 $$q + (\beta_1 - 1) q^{2} + (\beta_1 - 1) q^{3} - \beta_1 q^{4} + (\beta_1 - 1) q^{5} - \beta_1 q^{6} - \beta_{3} q^{7} + q^{8} - \beta_1 q^{9} - \beta_1 q^{10} + (\beta_{2} + 1) q^{11} + q^{12} + ( - \beta_{4} - \beta_{2}) q^{13} - \beta_{5} q^{14} - \beta_1 q^{15} + (\beta_1 - 1) q^{16} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 + 1) q^{17} + q^{18} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{19} + q^{20} - \beta_{5} q^{21} + (\beta_{4} + \beta_1 - 1) q^{22} + (\beta_{4} + \beta_{2} + \beta_1) q^{23} + (\beta_1 - 1) q^{24} - \beta_1 q^{25} + \beta_{2} q^{26} + q^{27} + (\beta_{5} + \beta_{3}) q^{28} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 3 \beta_1) q^{29}+ \cdots + ( - \beta_{4} - \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b1 - 1) * q^3 - b1 * q^4 + (b1 - 1) * q^5 - b1 * q^6 - b3 * q^7 + q^8 - b1 * q^9 - b1 * q^10 + (b2 + 1) * q^11 + q^12 + (-b4 - b2) * q^13 - b5 * q^14 - b1 * q^15 + (b1 - 1) * q^16 + (-b5 - 2*b4 - b1 + 1) * q^17 + q^18 + (-b5 - b4 - b2) * q^19 + q^20 - b5 * q^21 + (b4 + b1 - 1) * q^22 + (b4 + b2 + b1) * q^23 + (b1 - 1) * q^24 - b1 * q^25 + b2 * q^26 + q^27 + (b5 + b3) * q^28 + (-b5 + 2*b4 - b3 + 2*b2 - 3*b1) * q^29 + q^30 + b2 * q^31 - b1 * q^32 + (b4 + b1 - 1) * q^33 + (b5 + 2*b4 + b3 + 2*b2 + b1) * q^34 - b5 * q^35 + (b1 - 1) * q^36 - q^37 + (b5 + b3 + b2) * q^38 + b2 * q^39 + (b1 - 1) * q^40 + (-3*b5 - b4 + 2*b1 - 2) * q^41 + (b5 + b3) * q^42 + (2*b5 + b4 + 4*b1 - 4) * q^43 + (-b4 - b2 - b1) * q^44 + q^45 + (-b2 - 1) * q^46 + (-2*b5 - 2*b4 - 2*b3 - 2*b2) * q^47 - b1 * q^48 + (2*b3 + 2*b2 + 6) * q^49 + q^50 + (b5 + 2*b4 + b3 + 2*b2 + b1) * q^51 + b4 * q^52 + (b5 + b4 + b3 + b2 - 6*b1) * q^53 + (b1 - 1) * q^54 + (b4 + b1 - 1) * q^55 - b3 * q^56 + (b5 + b3 + b2) * q^57 + (b3 - 2*b2 + 3) * q^58 + (6*b1 - 6) * q^59 + (b1 - 1) * q^60 + (-b5 - b4 - b3 - b2 + b1) * q^61 + b4 * q^62 + (b5 + b3) * q^63 + q^64 + b2 * q^65 + (-b4 - b2 - b1) * q^66 + (-b5 - b4 - b3 - b2 - 5*b1) * q^67 + (-b3 - 2*b2 - 1) * q^68 + (-b2 - 1) * q^69 + (b5 + b3) * q^70 + (-b5 - 2*b4 - b1 + 1) * q^71 - b1 * q^72 + (b5 + b4 - 7*b1 + 7) * q^73 + (-b1 + 1) * q^74 + q^75 + (b4 - b3) * q^76 + (-b3 + 3*b2 + 4) * q^77 + b4 * q^78 + (-b4 - 2*b1 + 2) * q^79 - b1 * q^80 + (b1 - 1) * q^81 + (3*b5 + b4 + 3*b3 + b2 - 2*b1) * q^82 + (-4*b2 + 2) * q^83 - b3 * q^84 + (b5 + 2*b4 + b3 + 2*b2 + b1) * q^85 + (-2*b5 - b4 - 2*b3 - b2 - 4*b1) * q^86 + (b3 - 2*b2 + 3) * q^87 + (b2 + 1) * q^88 + (-b5 - b4 - b3 - b2 + 6*b1) * q^89 + (b1 - 1) * q^90 + (-3*b4 - 3*b2 - 4*b1) * q^91 + (-b4 - b1 + 1) * q^92 + b4 * q^93 + (2*b3 + 2*b2) * q^94 + (b5 + b3 + b2) * q^95 + q^96 + (b5 + 13*b1 - 13) * q^97 + (2*b5 + 2*b4 + 6*b1 - 6) * q^98 + (-b4 - b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 3 q^{6} + 2 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10})$$ 6 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 - 3 * q^5 - 3 * q^6 + 2 * q^7 + 6 * q^8 - 3 * q^9 $$6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 3 q^{6} + 2 q^{7} + 6 q^{8} - 3 q^{9} - 3 q^{10} + 8 q^{11} + 6 q^{12} - q^{13} - q^{14} - 3 q^{15} - 3 q^{16} + 4 q^{17} + 6 q^{18} - 2 q^{19} + 6 q^{20} - q^{21} - 4 q^{22} + 4 q^{23} - 3 q^{24} - 3 q^{25} + 2 q^{26} + 6 q^{27} - q^{28} - 6 q^{29} + 6 q^{30} + 2 q^{31} - 3 q^{32} - 4 q^{33} + 4 q^{34} - q^{35} - 3 q^{36} - 6 q^{37} + q^{38} + 2 q^{39} - 3 q^{40} - 8 q^{41} - q^{42} - 11 q^{43} - 4 q^{44} + 6 q^{45} - 8 q^{46} - 3 q^{48} + 36 q^{49} + 6 q^{50} + 4 q^{51} - q^{52} - 18 q^{53} - 3 q^{54} - 4 q^{55} + 2 q^{56} + q^{57} + 12 q^{58} - 18 q^{59} - 3 q^{60} + 3 q^{61} - q^{62} - q^{63} + 6 q^{64} + 2 q^{65} - 4 q^{66} - 15 q^{67} - 8 q^{68} - 8 q^{69} - q^{70} + 4 q^{71} - 3 q^{72} + 21 q^{73} + 3 q^{74} + 6 q^{75} + q^{76} + 32 q^{77} - q^{78} + 7 q^{79} - 3 q^{80} - 3 q^{81} - 8 q^{82} + 4 q^{83} + 2 q^{84} + 4 q^{85} - 11 q^{86} + 12 q^{87} + 8 q^{88} + 18 q^{89} - 3 q^{90} - 15 q^{91} + 4 q^{92} - q^{93} + q^{95} + 6 q^{96} - 38 q^{97} - 18 q^{98} - 4 q^{99}+O(q^{100})$$ 6 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 - 3 * q^5 - 3 * q^6 + 2 * q^7 + 6 * q^8 - 3 * q^9 - 3 * q^10 + 8 * q^11 + 6 * q^12 - q^13 - q^14 - 3 * q^15 - 3 * q^16 + 4 * q^17 + 6 * q^18 - 2 * q^19 + 6 * q^20 - q^21 - 4 * q^22 + 4 * q^23 - 3 * q^24 - 3 * q^25 + 2 * q^26 + 6 * q^27 - q^28 - 6 * q^29 + 6 * q^30 + 2 * q^31 - 3 * q^32 - 4 * q^33 + 4 * q^34 - q^35 - 3 * q^36 - 6 * q^37 + q^38 + 2 * q^39 - 3 * q^40 - 8 * q^41 - q^42 - 11 * q^43 - 4 * q^44 + 6 * q^45 - 8 * q^46 - 3 * q^48 + 36 * q^49 + 6 * q^50 + 4 * q^51 - q^52 - 18 * q^53 - 3 * q^54 - 4 * q^55 + 2 * q^56 + q^57 + 12 * q^58 - 18 * q^59 - 3 * q^60 + 3 * q^61 - q^62 - q^63 + 6 * q^64 + 2 * q^65 - 4 * q^66 - 15 * q^67 - 8 * q^68 - 8 * q^69 - q^70 + 4 * q^71 - 3 * q^72 + 21 * q^73 + 3 * q^74 + 6 * q^75 + q^76 + 32 * q^77 - q^78 + 7 * q^79 - 3 * q^80 - 3 * q^81 - 8 * q^82 + 4 * q^83 + 2 * q^84 + 4 * q^85 - 11 * q^86 + 12 * q^87 + 8 * q^88 + 18 * q^89 - 3 * q^90 - 15 * q^91 + 4 * q^92 - q^93 + q^95 + 6 * q^96 - 38 * q^97 - 18 * q^98 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 14x^{4} + 49x^{2} + 12$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 7\nu + 2 ) / 4$$ (v^3 + 7*v + 2) / 4 $$\beta_{2}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} - 9\nu^{2} - 10 ) / 2$$ (-v^4 - 9*v^2 - 10) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 12\nu^{3} - 2\nu^{2} + 35\nu - 10 ) / 4$$ (v^5 + 12*v^3 - 2*v^2 + 35*v - 10) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 12\nu^{3} + 9\nu^{2} - 29\nu + 10 ) / 4$$ (-v^5 + v^4 - 12*v^3 + 9*v^2 - 29*v + 10) / 4
 $$\nu$$ $$=$$ $$( 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 3$$ (2*b5 + 2*b4 + b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 5$$ b2 - 5 $$\nu^{3}$$ $$=$$ $$( -14\beta_{5} - 14\beta_{4} - 7\beta_{3} - 7\beta_{2} + 12\beta _1 - 6 ) / 3$$ (-14*b5 - 14*b4 - 7*b3 - 7*b2 + 12*b1 - 6) / 3 $$\nu^{4}$$ $$=$$ $$-2\beta_{3} - 9\beta_{2} + 35$$ -2*b3 - 9*b2 + 35 $$\nu^{5}$$ $$=$$ $$( 98\beta_{5} + 110\beta_{4} + 49\beta_{3} + 55\beta_{2} - 144\beta _1 + 72 ) / 3$$ (98*b5 + 110*b4 + 49*b3 + 55*b2 - 144*b1 + 72) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 − 2.35084i 2.86514i − 0.514306i 2.35084i − 2.86514i 0.514306i
−0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −4.59821 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
121.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.75353 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
121.3 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 3.84469 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
391.1 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −4.59821 1.00000 −0.500000 0.866025i −0.500000 0.866025i
391.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.75353 1.00000 −0.500000 0.866025i −0.500000 0.866025i
391.3 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 3.84469 1.00000 −0.500000 0.866025i −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.i.j 6
3.b odd 2 1 1710.2.l.q 6
19.c even 3 1 inner 570.2.i.j 6
57.h odd 6 1 1710.2.l.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.j 6 1.a even 1 1 trivial
570.2.i.j 6 19.c even 3 1 inner
1710.2.l.q 6 3.b odd 2 1
1710.2.l.q 6 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$:

 $$T_{7}^{3} - T_{7}^{2} - 19T_{7} + 31$$ T7^3 - T7^2 - 19*T7 + 31 $$T_{11}^{3} - 4T_{11}^{2} - 11T_{11} + 6$$ T11^3 - 4*T11^2 - 11*T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{3}$$
$3$ $$(T^{2} + T + 1)^{3}$$
$5$ $$(T^{2} + T + 1)^{3}$$
$7$ $$(T^{3} - T^{2} - 19 T + 31)^{2}$$
$11$ $$(T^{3} - 4 T^{2} - 11 T + 6)^{2}$$
$13$ $$T^{6} + T^{5} + \cdots + 64$$
$17$ $$T^{6} - 4 T^{5} + \cdots + 46656$$
$19$ $$T^{6} + 2 T^{5} + \cdots + 6859$$
$23$ $$T^{6} - 4 T^{5} + \cdots + 36$$
$29$ $$T^{6} + 6 T^{5} + \cdots + 467856$$
$31$ $$(T^{3} - T^{2} - 16 T - 8)^{2}$$
$37$ $$(T + 1)^{6}$$
$41$ $$T^{6} + 8 T^{5} + \cdots + 1077444$$
$43$ $$T^{6} + 11 T^{5} + \cdots + 1296$$
$47$ $$T^{6} + 84 T^{4} + \cdots + 20736$$
$53$ $$T^{6} + 18 T^{5} + \cdots + 11664$$
$59$ $$(T^{2} + 6 T + 36)^{3}$$
$61$ $$T^{6} - 3 T^{5} + \cdots + 4$$
$67$ $$T^{6} + 15 T^{5} + \cdots + 4$$
$71$ $$T^{6} - 4 T^{5} + \cdots + 46656$$
$73$ $$T^{6} - 21 T^{5} + \cdots + 45796$$
$79$ $$T^{6} - 7 T^{5} + \cdots + 144$$
$83$ $$(T^{3} - 2 T^{2} + \cdots + 1032)^{2}$$
$89$ $$T^{6} - 18 T^{5} + \cdots + 11664$$
$97$ $$T^{6} + 38 T^{5} + \cdots + 3283344$$