Properties

 Label 5202.2.a.g Level $5202$ Weight $2$ Character orbit 5202.a Self dual yes Analytic conductor $41.538$ Analytic rank $1$ 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] = [5202,2,Mod(1,5202)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5202, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5202.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5202 = 2 \cdot 3^{2} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5202.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: $$41.5381791315$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 4 q^{5} + 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - 4 * q^5 + 2 * q^7 + q^8 $$q + q^{2} + q^{4} - 4 q^{5} + 2 q^{7} + q^{8} - 4 q^{10} - 6 q^{13} + 2 q^{14} + q^{16} + 4 q^{19} - 4 q^{20} + 6 q^{23} + 11 q^{25} - 6 q^{26} + 2 q^{28} - 4 q^{29} + 6 q^{31} + q^{32} - 8 q^{35} + 4 q^{37} + 4 q^{38} - 4 q^{40} - 10 q^{41} - 4 q^{43} + 6 q^{46} - 4 q^{47} - 3 q^{49} + 11 q^{50} - 6 q^{52} + 2 q^{53} + 2 q^{56} - 4 q^{58} - 12 q^{59} + 4 q^{61} + 6 q^{62} + q^{64} + 24 q^{65} - 12 q^{67} - 8 q^{70} - 6 q^{71} - 2 q^{73} + 4 q^{74} + 4 q^{76} - 10 q^{79} - 4 q^{80} - 10 q^{82} + 12 q^{83} - 4 q^{86} + 2 q^{89} - 12 q^{91} + 6 q^{92} - 4 q^{94} - 16 q^{95} - 6 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 - 4 * q^5 + 2 * q^7 + q^8 - 4 * q^10 - 6 * q^13 + 2 * q^14 + q^16 + 4 * q^19 - 4 * q^20 + 6 * q^23 + 11 * q^25 - 6 * q^26 + 2 * q^28 - 4 * q^29 + 6 * q^31 + q^32 - 8 * q^35 + 4 * q^37 + 4 * q^38 - 4 * q^40 - 10 * q^41 - 4 * q^43 + 6 * q^46 - 4 * q^47 - 3 * q^49 + 11 * q^50 - 6 * q^52 + 2 * q^53 + 2 * q^56 - 4 * q^58 - 12 * q^59 + 4 * q^61 + 6 * q^62 + q^64 + 24 * q^65 - 12 * q^67 - 8 * q^70 - 6 * q^71 - 2 * q^73 + 4 * q^74 + 4 * q^76 - 10 * q^79 - 4 * q^80 - 10 * q^82 + 12 * q^83 - 4 * q^86 + 2 * q^89 - 12 * q^91 + 6 * q^92 - 4 * q^94 - 16 * q^95 - 6 * 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 0 1.00000 −4.00000 0 2.00000 1.00000 0 −4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$17$$ $$1$$

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 5202.2.a.g 1
3.b odd 2 1 1734.2.a.h 1
17.b even 2 1 306.2.a.d 1
51.c odd 2 1 102.2.a.a 1
51.f odd 4 2 1734.2.b.d 2
51.g odd 8 4 1734.2.f.g 4
68.d odd 2 1 2448.2.a.t 1
85.c even 2 1 7650.2.a.z 1
136.e odd 2 1 9792.2.a.b 1
136.h even 2 1 9792.2.a.a 1
204.h even 2 1 816.2.a.h 1
255.h odd 2 1 2550.2.a.be 1
255.o even 4 2 2550.2.d.q 2
357.c even 2 1 4998.2.a.x 1
408.b odd 2 1 3264.2.a.bf 1
408.h even 2 1 3264.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.a 1 51.c odd 2 1
306.2.a.d 1 17.b even 2 1
816.2.a.h 1 204.h even 2 1
1734.2.a.h 1 3.b odd 2 1
1734.2.b.d 2 51.f odd 4 2
1734.2.f.g 4 51.g odd 8 4
2448.2.a.t 1 68.d odd 2 1
2550.2.a.be 1 255.h odd 2 1
2550.2.d.q 2 255.o even 4 2
3264.2.a.p 1 408.h even 2 1
3264.2.a.bf 1 408.b odd 2 1
4998.2.a.x 1 357.c even 2 1
5202.2.a.g 1 1.a even 1 1 trivial
7650.2.a.z 1 85.c even 2 1
9792.2.a.a 1 136.h even 2 1
9792.2.a.b 1 136.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5202))$$:

 $$T_{5} + 4$$ T5 + 4 $$T_{7} - 2$$ T7 - 2 $$T_{23} - 6$$ T23 - 6 $$T_{47} + 4$$ T47 + 4

Hecke characteristic polynomials

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