Properties

 Label 285.2.i.a Level $285$ Weight $2$ Character orbit 285.i Analytic conductor $2.276$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(106,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("285.106");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.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: $$2.27573645761$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/285\mathbb{Z}\right)^\times$$.

 $$n$$ $$172$$ $$191$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i −0.500000 + 0.866025i −1.00000 1.73205i 0.500000 0.866025i −1.00000 1.73205i −2.00000 0 −0.500000 0.866025i 1.00000 + 1.73205i
121.1 −1.00000 1.73205i −0.500000 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i −1.00000 + 1.73205i −2.00000 0 −0.500000 + 0.866025i 1.00000 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 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 285.2.i.a 2
3.b odd 2 1 855.2.k.d 2
19.c even 3 1 inner 285.2.i.a 2
19.c even 3 1 5415.2.a.k 1
19.d odd 6 1 5415.2.a.a 1
57.h odd 6 1 855.2.k.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.a 2 1.a even 1 1 trivial
285.2.i.a 2 19.c even 3 1 inner
855.2.k.d 2 3.b odd 2 1
855.2.k.d 2 57.h odd 6 1
5415.2.a.a 1 19.d odd 6 1
5415.2.a.k 1 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(285, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$(T + 2)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 6T + 36$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$T^{2} + 7T + 49$$
$31$ $$(T - 9)^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} + 10T + 100$$
$47$ $$T^{2} + 4T + 16$$
$53$ $$T^{2} + 14T + 196$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2} + 7T + 49$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} - 12T + 144$$