Properties

 Label 570.2.u.h Level $570$ Weight $2$ Character orbit 570.u 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(61,570)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(570, base_ring=CyclotomicField(18))

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("570.61");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.u (of order $$9$$, degree $$6$$, 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: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

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

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-\zeta_{18}^{5}$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 −0.173648 − 0.984808i −0.173648 + 0.984808i 0.939693 − 0.342020i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.766044 + 0.642788i
−0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i −0.766044 + 0.642788i 0.173648 0.984808i 2.43969 4.22567i −0.500000 0.866025i −0.939693 + 0.342020i 0.939693 0.342020i
271.1 −0.939693 + 0.342020i 0.173648 0.984808i 0.766044 0.642788i −0.766044 0.642788i 0.173648 + 0.984808i 2.43969 + 4.22567i −0.500000 + 0.866025i −0.939693 0.342020i 0.939693 + 0.342020i
301.1 0.766044 + 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.173648 + 0.984808i −0.939693 0.342020i 0.733956 + 1.27125i −0.500000 + 0.866025i 0.766044 0.642788i −0.766044 + 0.642788i
481.1 0.766044 0.642788i −0.939693 0.342020i 0.173648 0.984808i −0.173648 0.984808i −0.939693 + 0.342020i 0.733956 1.27125i −0.500000 0.866025i 0.766044 + 0.642788i −0.766044 0.642788i
511.1 0.173648 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 0.939693 0.342020i 0.766044 0.642788i 1.32635 + 2.29731i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.173648 0.984808i
541.1 0.173648 + 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 + 0.642788i 1.32635 2.29731i −0.500000 0.866025i 0.173648 0.984808i −0.173648 + 0.984808i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.1 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.e even 9 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.u.h 6
19.e even 9 1 inner 570.2.u.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.h 6 1.a even 1 1 trivial
570.2.u.h 6 19.e even 9 1 inner

Hecke kernels

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

 $$T_{7}^{6} - 9T_{7}^{5} + 57T_{7}^{4} - 178T_{7}^{3} + 405T_{7}^{2} - 456T_{7} + 361$$ T7^6 - 9*T7^5 + 57*T7^4 - 178*T7^3 + 405*T7^2 - 456*T7 + 361 $$T_{11}^{6} + 9T_{11}^{5} + 57T_{11}^{4} + 182T_{11}^{3} + 423T_{11}^{2} + 408T_{11} + 289$$ T11^6 + 9*T11^5 + 57*T11^4 + 182*T11^3 + 423*T11^2 + 408*T11 + 289

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6} + T^{3} + 1$$
$5$ $$T^{6} - T^{3} + 1$$
$7$ $$T^{6} - 9 T^{5} + \cdots + 361$$
$11$ $$T^{6} + 9 T^{5} + \cdots + 289$$
$13$ $$T^{6} - 9 T^{5} + \cdots + 81$$
$17$ $$T^{6} + 9 T^{5} + \cdots + 361$$
$19$ $$T^{6} + 9 T^{5} + \cdots + 6859$$
$23$ $$T^{6} - 3 T^{5} + \cdots + 3249$$
$29$ $$T^{6} + 9 T^{5} + \cdots + 1$$
$31$ $$T^{6} - 18 T^{5} + \cdots + 83521$$
$37$ $$(T^{3} + 3 T^{2} + \cdots - 163)^{2}$$
$41$ $$T^{6} + 6 T^{5} + \cdots + 11881$$
$43$ $$T^{6} - 21 T^{5} + \cdots + 1369$$
$47$ $$T^{6} + 36 T^{4} + \cdots + 81$$
$53$ $$T^{6} - 12 T^{5} + \cdots + 25281$$
$59$ $$T^{6} - 9 T^{5} + \cdots + 1$$
$61$ $$T^{6} - 3 T^{5} + \cdots + 72361$$
$67$ $$T^{6} - 36 T^{5} + \cdots + 908209$$
$71$ $$T^{6} + 36 T^{5} + \cdots + 641601$$
$73$ $$T^{6} - 21 T^{5} + \cdots + 1819801$$
$79$ $$T^{6} + 9 T^{5} + \cdots + 5041$$
$83$ $$T^{6} + 3 T^{5} + \cdots + 5329$$
$89$ $$T^{6} - 3 T^{5} + \cdots + 47961$$
$97$ $$T^{6} + 343 T^{3} + 117649$$