# Properties

 Label 285.2.i.d 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 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.

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

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

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

## Character values

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

 $$n$$ $$172$$ $$191$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−1.20711 + 2.09077i 0.500000 0.866025i −1.91421 3.31552i 0.500000 0.866025i 1.20711 + 2.09077i −3.82843 4.41421 −0.500000 0.866025i 1.20711 + 2.09077i
106.2 0.207107 0.358719i 0.500000 0.866025i 0.914214 + 1.58346i 0.500000 0.866025i −0.207107 0.358719i 1.82843 1.58579 −0.500000 0.866025i −0.207107 0.358719i
121.1 −1.20711 2.09077i 0.500000 + 0.866025i −1.91421 + 3.31552i 0.500000 + 0.866025i 1.20711 2.09077i −3.82843 4.41421 −0.500000 + 0.866025i 1.20711 2.09077i
121.2 0.207107 + 0.358719i 0.500000 + 0.866025i 0.914214 1.58346i 0.500000 + 0.866025i −0.207107 + 0.358719i 1.82843 1.58579 −0.500000 + 0.866025i −0.207107 + 0.358719i
 $$n$$: 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.d 4
3.b odd 2 1 855.2.k.f 4
19.c even 3 1 inner 285.2.i.d 4
19.c even 3 1 5415.2.a.u 2
19.d odd 6 1 5415.2.a.o 2
57.h odd 6 1 855.2.k.f 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(285, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$(T^{2} + 2 T - 7)^{2}$$
$11$ $$(T^{2} - 8)^{2}$$
$13$ $$T^{4} + 2 T^{3} + 11 T^{2} - 14 T + 49$$
$17$ $$T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64$$
$19$ $$(T^{2} - 8 T + 19)^{2}$$
$23$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$29$ $$T^{4} + 8 T^{3} + 80 T^{2} - 128 T + 256$$
$31$ $$(T + 5)^{4}$$
$37$ $$(T^{2} + 10 T + 17)^{2}$$
$41$ $$T^{4} + 8T^{2} + 64$$
$43$ $$T^{4} + 10 T^{3} + 83 T^{2} + \cdots + 289$$
$47$ $$T^{4} - 12 T^{3} + 116 T^{2} + \cdots + 784$$
$53$ $$(T^{2} - 2 T + 4)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 6 T^{3} + 155 T^{2} + \cdots + 14161$$
$67$ $$T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969$$
$71$ $$(T^{2} + 10 T + 100)^{2}$$
$73$ $$T^{4} + 2 T^{3} + 75 T^{2} + \cdots + 5041$$
$79$ $$T^{4} - 18 T^{3} + 275 T^{2} + \cdots + 2401$$
$83$ $$(T + 8)^{4}$$
$89$ $$T^{4} + 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$97$ $$(T^{2} - 6 T + 36)^{2}$$