# Properties

 Label 285.2.i.e Level $285$ Weight $2$ Character orbit 285.i Analytic conductor $2.276$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 1 ) / 2$$ (v^3 + 1) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_1$$ b3 + b2 + b1 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 1$$ 2*b2 - 1

## Character values

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

 $$n$$ $$172$$ $$191$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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
 −0.309017 + 0.535233i 0.809017 − 1.40126i −0.309017 − 0.535233i 0.809017 + 1.40126i
−0.309017 + 0.535233i −0.500000 + 0.866025i 0.809017 + 1.40126i 0.500000 0.866025i −0.309017 0.535233i 5.23607 −2.23607 −0.500000 0.866025i 0.309017 + 0.535233i
106.2 0.809017 1.40126i −0.500000 + 0.866025i −0.309017 0.535233i 0.500000 0.866025i 0.809017 + 1.40126i 0.763932 2.23607 −0.500000 0.866025i −0.809017 1.40126i
121.1 −0.309017 0.535233i −0.500000 0.866025i 0.809017 1.40126i 0.500000 + 0.866025i −0.309017 + 0.535233i 5.23607 −2.23607 −0.500000 + 0.866025i 0.309017 0.535233i
121.2 0.809017 + 1.40126i −0.500000 0.866025i −0.309017 + 0.535233i 0.500000 + 0.866025i 0.809017 1.40126i 0.763932 2.23607 −0.500000 + 0.866025i −0.809017 + 1.40126i
 $$n$$: 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.e 4
3.b odd 2 1 855.2.k.e 4
19.c even 3 1 inner 285.2.i.e 4
19.c even 3 1 5415.2.a.q 2
19.d odd 6 1 5415.2.a.t 2
57.h odd 6 1 855.2.k.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.e 4 1.a even 1 1 trivial
285.2.i.e 4 19.c even 3 1 inner
855.2.k.e 4 3.b odd 2 1
855.2.k.e 4 57.h odd 6 1
5415.2.a.q 2 19.c even 3 1
5415.2.a.t 2 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} + 2T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(285, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + 2 T^{2} + T + 1$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$(T^{2} - 6 T + 4)^{2}$$
$11$ $$(T - 2)^{4}$$
$13$ $$T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16$$
$17$ $$T^{4} + 2 T^{3} + 23 T^{2} - 38 T + 361$$
$19$ $$T^{4} + 4 T^{3} - 3 T^{2} + 76 T + 361$$
$23$ $$T^{4} + 4 T^{3} + 32 T^{2} - 64 T + 256$$
$29$ $$T^{4} - 6 T^{3} + 32 T^{2} - 24 T + 16$$
$31$ $$(T^{2} + 12 T + 31)^{2}$$
$37$ $$(T^{2} + 4 T - 16)^{2}$$
$41$ $$T^{4} + 2 T^{3} + 48 T^{2} + \cdots + 1936$$
$43$ $$T^{4} + 10 T^{3} + 120 T^{2} + \cdots + 400$$
$47$ $$T^{4} + 8 T^{3} + 53 T^{2} + 88 T + 121$$
$53$ $$(T^{2} - T + 1)^{2}$$
$59$ $$(T^{2} - 2 T + 4)^{2}$$
$61$ $$T^{4} + 16 T^{3} + 212 T^{2} + \cdots + 1936$$
$67$ $$T^{4} - 2 T^{3} + 8 T^{2} + 8 T + 16$$
$71$ $$T^{4} + 14 T^{3} + 152 T^{2} + \cdots + 1936$$
$73$ $$T^{4} - 16 T^{3} + 212 T^{2} + \cdots + 1936$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 8 T - 109)^{2}$$
$89$ $$T^{4} + 16 T^{3} + 212 T^{2} + \cdots + 1936$$
$97$ $$(T^{2} - 2 T + 4)^{2}$$