Properties

 Label 390.2.a.d Level $390$ Weight $2$ Character orbit 390.a Self dual yes Analytic conductor $3.114$ 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] = [390,2,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.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: $$3.11416567883$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 2 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 + 2 * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 2 q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + q^{13} - 2 q^{14} + q^{15} + q^{16} - q^{18} + 2 q^{19} + q^{20} + 2 q^{21} - 6 q^{23} - q^{24} + q^{25} - q^{26} + q^{27} + 2 q^{28} - q^{30} + 8 q^{31} - q^{32} + 2 q^{35} + q^{36} + 2 q^{37} - 2 q^{38} + q^{39} - q^{40} + 6 q^{41} - 2 q^{42} - 4 q^{43} + q^{45} + 6 q^{46} + q^{48} - 3 q^{49} - q^{50} + q^{52} - 6 q^{53} - q^{54} - 2 q^{56} + 2 q^{57} + q^{60} + 14 q^{61} - 8 q^{62} + 2 q^{63} + q^{64} + q^{65} - 4 q^{67} - 6 q^{69} - 2 q^{70} - q^{72} - 4 q^{73} - 2 q^{74} + q^{75} + 2 q^{76} - q^{78} - 16 q^{79} + q^{80} + q^{81} - 6 q^{82} - 12 q^{83} + 2 q^{84} + 4 q^{86} - 6 q^{89} - q^{90} + 2 q^{91} - 6 q^{92} + 8 q^{93} + 2 q^{95} - q^{96} - 4 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 + 2 * q^7 - q^8 + q^9 - q^10 + q^12 + q^13 - 2 * q^14 + q^15 + q^16 - q^18 + 2 * q^19 + q^20 + 2 * q^21 - 6 * q^23 - q^24 + q^25 - q^26 + q^27 + 2 * q^28 - q^30 + 8 * q^31 - q^32 + 2 * q^35 + q^36 + 2 * q^37 - 2 * q^38 + q^39 - q^40 + 6 * q^41 - 2 * q^42 - 4 * q^43 + q^45 + 6 * q^46 + q^48 - 3 * q^49 - q^50 + q^52 - 6 * q^53 - q^54 - 2 * q^56 + 2 * q^57 + q^60 + 14 * q^61 - 8 * q^62 + 2 * q^63 + q^64 + q^65 - 4 * q^67 - 6 * q^69 - 2 * q^70 - q^72 - 4 * q^73 - 2 * q^74 + q^75 + 2 * q^76 - q^78 - 16 * q^79 + q^80 + q^81 - 6 * q^82 - 12 * q^83 + 2 * q^84 + 4 * q^86 - 6 * q^89 - q^90 + 2 * q^91 - 6 * q^92 + 8 * q^93 + 2 * q^95 - q^96 - 4 * q^97 + 3 * q^98

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 2.00000 −1.00000 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

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 390.2.a.d 1
3.b odd 2 1 1170.2.a.k 1
4.b odd 2 1 3120.2.a.j 1
5.b even 2 1 1950.2.a.o 1
5.c odd 4 2 1950.2.e.d 2
12.b even 2 1 9360.2.a.g 1
13.b even 2 1 5070.2.a.t 1
13.d odd 4 2 5070.2.b.m 2
15.d odd 2 1 5850.2.a.g 1
15.e even 4 2 5850.2.e.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.d 1 1.a even 1 1 trivial
1170.2.a.k 1 3.b odd 2 1
1950.2.a.o 1 5.b even 2 1
1950.2.e.d 2 5.c odd 4 2
3120.2.a.j 1 4.b odd 2 1
5070.2.a.t 1 13.b even 2 1
5070.2.b.m 2 13.d odd 4 2
5850.2.a.g 1 15.d odd 2 1
5850.2.e.o 2 15.e even 4 2
9360.2.a.g 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(390))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11}$$ T11 $$T_{31} - 8$$ T31 - 8

Hecke characteristic polynomials

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