# Properties

 Label 285.2.k.a Level $285$ Weight $2$ Character orbit 285.k Analytic conductor $2.276$ Analytic rank $1$ 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(77,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("285.77");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.k (of order $$4$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$172$$ $$191$$ $$211$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
−1.00000 + 1.00000i 1.73205i 0 −0.133975 + 2.23205i 1.73205 + 1.73205i −2.36603 2.36603i −2.00000 2.00000i −3.00000 −2.09808 2.36603i
77.2 −1.00000 + 1.00000i 1.73205i 0 −1.86603 1.23205i −1.73205 1.73205i −0.633975 0.633975i −2.00000 2.00000i −3.00000 3.09808 0.633975i
248.1 −1.00000 1.00000i 1.73205i 0 −1.86603 + 1.23205i −1.73205 + 1.73205i −0.633975 + 0.633975i −2.00000 + 2.00000i −3.00000 3.09808 + 0.633975i
248.2 −1.00000 1.00000i 1.73205i 0 −0.133975 2.23205i 1.73205 1.73205i −2.36603 + 2.36603i −2.00000 + 2.00000i −3.00000 −2.09808 + 2.36603i
 $$n$$: 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
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.k.a 4
3.b odd 2 1 285.2.k.b yes 4
5.c odd 4 1 285.2.k.b yes 4
15.e even 4 1 inner 285.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.k.a 4 1.a even 1 1 trivial
285.2.k.a 4 15.e even 4 1 inner
285.2.k.b yes 4 3.b odd 2 1
285.2.k.b yes 4 5.c odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 2)^{2}$$
$3$ $$(T^{2} + 3)^{2}$$
$5$ $$T^{4} + 4 T^{3} + \cdots + 25$$
$7$ $$T^{4} + 6 T^{3} + \cdots + 9$$
$11$ $$T^{4} + 14T^{2} + 1$$
$13$ $$T^{4} + 8 T^{3} + \cdots + 4$$
$17$ $$T^{4} + 10 T^{3} + \cdots + 1$$
$19$ $$(T^{2} + 1)^{2}$$
$23$ $$T^{4} + 12 T^{3} + \cdots + 36$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 14 T + 46)^{2}$$
$37$ $$T^{4} + 4 T^{3} + \cdots + 484$$
$41$ $$T^{4} + 152T^{2} + 5476$$
$43$ $$T^{4} + 10 T^{3} + \cdots + 1$$
$47$ $$T^{4} + 10 T^{3} + \cdots + 3721$$
$53$ $$T^{4} - 16 T^{3} + \cdots + 676$$
$59$ $$(T^{2} - 10 T - 2)^{2}$$
$61$ $$(T^{2} + 12 T + 33)^{2}$$
$67$ $$T^{4} - 12 T^{3} + \cdots + 1296$$
$71$ $$T^{4} + 168T^{2} + 6084$$
$73$ $$T^{4} - 18 T^{3} + \cdots + 1521$$
$79$ $$(T^{2} + 108)^{2}$$
$83$ $$T^{4} + 8 T^{3} + \cdots + 4$$
$89$ $$(T^{2} + 18 T + 78)^{2}$$
$97$ $$T^{4} + 12 T^{3} + \cdots + 36$$