Properties

 Label 285.2.a.c Level $285$ Weight $2$ Character orbit 285.a Self dual yes Analytic conductor $2.276$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.a (trivial)

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: yes Analytic conductor: $$2.27573645761$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 −1.00000 −1.00000 1.00000 −1.00000 4.00000 −3.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.a.c 1
3.b odd 2 1 855.2.a.a 1
4.b odd 2 1 4560.2.a.w 1
5.b even 2 1 1425.2.a.c 1
5.c odd 4 2 1425.2.c.f 2
15.d odd 2 1 4275.2.a.j 1
19.b odd 2 1 5415.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.c 1 1.a even 1 1 trivial
855.2.a.a 1 3.b odd 2 1
1425.2.a.c 1 5.b even 2 1
1425.2.c.f 2 5.c odd 4 2
4275.2.a.j 1 15.d odd 2 1
4560.2.a.w 1 4.b odd 2 1
5415.2.a.e 1 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(285))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} - 4$$ T7 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T - 4$$
$11$ $$T - 4$$
$13$ $$T - 2$$
$17$ $$T - 2$$
$19$ $$T + 1$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T + 6$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T + 12$$
$53$ $$T + 14$$
$59$ $$T - 4$$
$61$ $$T - 14$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T - 16$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 10$$