Properties

 Label 5390.2.a.x Level $5390$ Weight $2$ Character orbit 5390.a Self dual yes Analytic conductor $43.039$ 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] = [5390,2,Mod(1,5390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5390, 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("5390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5390.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: $$43.0393666895$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + q^8 - 2 * q^9 $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} - 2 q^{9} + q^{10} - q^{11} - q^{12} - 2 q^{13} - q^{15} + q^{16} + 3 q^{17} - 2 q^{18} + q^{19} + q^{20} - q^{22} + 6 q^{23} - q^{24} + q^{25} - 2 q^{26} + 5 q^{27} - 9 q^{29} - q^{30} - 5 q^{31} + q^{32} + q^{33} + 3 q^{34} - 2 q^{36} + 5 q^{37} + q^{38} + 2 q^{39} + q^{40} + 6 q^{41} + 8 q^{43} - q^{44} - 2 q^{45} + 6 q^{46} - 6 q^{47} - q^{48} + q^{50} - 3 q^{51} - 2 q^{52} + 9 q^{53} + 5 q^{54} - q^{55} - q^{57} - 9 q^{58} - 6 q^{59} - q^{60} - 5 q^{61} - 5 q^{62} + q^{64} - 2 q^{65} + q^{66} + 8 q^{67} + 3 q^{68} - 6 q^{69} - 9 q^{71} - 2 q^{72} + 10 q^{73} + 5 q^{74} - q^{75} + q^{76} + 2 q^{78} + 14 q^{79} + q^{80} + q^{81} + 6 q^{82} + 6 q^{83} + 3 q^{85} + 8 q^{86} + 9 q^{87} - q^{88} + 15 q^{89} - 2 q^{90} + 6 q^{92} + 5 q^{93} - 6 q^{94} + q^{95} - q^{96} - 8 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + q^8 - 2 * q^9 + q^10 - q^11 - q^12 - 2 * q^13 - q^15 + q^16 + 3 * q^17 - 2 * q^18 + q^19 + q^20 - q^22 + 6 * q^23 - q^24 + q^25 - 2 * q^26 + 5 * q^27 - 9 * q^29 - q^30 - 5 * q^31 + q^32 + q^33 + 3 * q^34 - 2 * q^36 + 5 * q^37 + q^38 + 2 * q^39 + q^40 + 6 * q^41 + 8 * q^43 - q^44 - 2 * q^45 + 6 * q^46 - 6 * q^47 - q^48 + q^50 - 3 * q^51 - 2 * q^52 + 9 * q^53 + 5 * q^54 - q^55 - q^57 - 9 * q^58 - 6 * q^59 - q^60 - 5 * q^61 - 5 * q^62 + q^64 - 2 * q^65 + q^66 + 8 * q^67 + 3 * q^68 - 6 * q^69 - 9 * q^71 - 2 * q^72 + 10 * q^73 + 5 * q^74 - q^75 + q^76 + 2 * q^78 + 14 * q^79 + q^80 + q^81 + 6 * q^82 + 6 * q^83 + 3 * q^85 + 8 * q^86 + 9 * q^87 - q^88 + 15 * q^89 - 2 * q^90 + 6 * q^92 + 5 * q^93 - 6 * q^94 + q^95 - q^96 - 8 * q^97 + 2 * 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 0 1.00000 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$+1$$

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 5390.2.a.x 1
7.b odd 2 1 110.2.a.c 1
21.c even 2 1 990.2.a.f 1
28.d even 2 1 880.2.a.d 1
35.c odd 2 1 550.2.a.d 1
35.f even 4 2 550.2.b.c 2
56.e even 2 1 3520.2.a.ba 1
56.h odd 2 1 3520.2.a.k 1
77.b even 2 1 1210.2.a.e 1
84.h odd 2 1 7920.2.a.bc 1
105.g even 2 1 4950.2.a.bm 1
105.k odd 4 2 4950.2.c.s 2
140.c even 2 1 4400.2.a.t 1
140.j odd 4 2 4400.2.b.j 2
308.g odd 2 1 9680.2.a.g 1
385.h even 2 1 6050.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.c 1 7.b odd 2 1
550.2.a.d 1 35.c odd 2 1
550.2.b.c 2 35.f even 4 2
880.2.a.d 1 28.d even 2 1
990.2.a.f 1 21.c even 2 1
1210.2.a.e 1 77.b even 2 1
3520.2.a.k 1 56.h odd 2 1
3520.2.a.ba 1 56.e even 2 1
4400.2.a.t 1 140.c even 2 1
4400.2.b.j 2 140.j odd 4 2
4950.2.a.bm 1 105.g even 2 1
4950.2.c.s 2 105.k odd 4 2
5390.2.a.x 1 1.a even 1 1 trivial
6050.2.a.bc 1 385.h even 2 1
7920.2.a.bc 1 84.h odd 2 1
9680.2.a.g 1 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5390))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 3$$ T17 - 3 $$T_{19} - 1$$ T19 - 1 $$T_{31} + 5$$ T31 + 5

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T - 3$$
$19$ $$T - 1$$
$23$ $$T - 6$$
$29$ $$T + 9$$
$31$ $$T + 5$$
$37$ $$T - 5$$
$41$ $$T - 6$$
$43$ $$T - 8$$
$47$ $$T + 6$$
$53$ $$T - 9$$
$59$ $$T + 6$$
$61$ $$T + 5$$
$67$ $$T - 8$$
$71$ $$T + 9$$
$73$ $$T - 10$$
$79$ $$T - 14$$
$83$ $$T - 6$$
$89$ $$T - 15$$
$97$ $$T + 8$$