Properties

 Label 9200.2.a.bj Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ 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] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.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: $$73.4623698596$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 184) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 - 2 * q^7 + 6 * q^9 $$q + 3 q^{3} - 2 q^{7} + 6 q^{9} + 5 q^{13} + 6 q^{17} - 6 q^{19} - 6 q^{21} + q^{23} + 9 q^{27} + 9 q^{29} - 3 q^{31} + 8 q^{37} + 15 q^{39} + 3 q^{41} - 8 q^{43} + 7 q^{47} - 3 q^{49} + 18 q^{51} + 2 q^{53} - 18 q^{57} - 4 q^{59} - 10 q^{61} - 12 q^{63} + 8 q^{67} + 3 q^{69} - 7 q^{71} - 9 q^{73} + 6 q^{79} + 9 q^{81} - 14 q^{83} + 27 q^{87} + 16 q^{89} - 10 q^{91} - 9 q^{93} - 6 q^{97}+O(q^{100})$$ q + 3 * q^3 - 2 * q^7 + 6 * q^9 + 5 * q^13 + 6 * q^17 - 6 * q^19 - 6 * q^21 + q^23 + 9 * q^27 + 9 * q^29 - 3 * q^31 + 8 * q^37 + 15 * q^39 + 3 * q^41 - 8 * q^43 + 7 * q^47 - 3 * q^49 + 18 * q^51 + 2 * q^53 - 18 * q^57 - 4 * q^59 - 10 * q^61 - 12 * q^63 + 8 * q^67 + 3 * q^69 - 7 * q^71 - 9 * q^73 + 6 * q^79 + 9 * q^81 - 14 * q^83 + 27 * q^87 + 16 * q^89 - 10 * q^91 - 9 * q^93 - 6 * q^97

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
0 3.00000 0 0 0 −2.00000 0 6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$-1$$

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 9200.2.a.bj 1
4.b odd 2 1 4600.2.a.a 1
5.b even 2 1 368.2.a.a 1
15.d odd 2 1 3312.2.a.i 1
20.d odd 2 1 184.2.a.d 1
20.e even 4 2 4600.2.e.a 2
40.e odd 2 1 1472.2.a.a 1
40.f even 2 1 1472.2.a.m 1
60.h even 2 1 1656.2.a.c 1
115.c odd 2 1 8464.2.a.b 1
140.c even 2 1 9016.2.a.b 1
460.g even 2 1 4232.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.d 1 20.d odd 2 1
368.2.a.a 1 5.b even 2 1
1472.2.a.a 1 40.e odd 2 1
1472.2.a.m 1 40.f even 2 1
1656.2.a.c 1 60.h even 2 1
3312.2.a.i 1 15.d odd 2 1
4232.2.a.j 1 460.g even 2 1
4600.2.a.a 1 4.b odd 2 1
4600.2.e.a 2 20.e even 4 2
8464.2.a.b 1 115.c odd 2 1
9016.2.a.b 1 140.c even 2 1
9200.2.a.bj 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{7} + 2$$ T7 + 2 $$T_{11}$$ T11

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T - 5$$
$17$ $$T - 6$$
$19$ $$T + 6$$
$23$ $$T - 1$$
$29$ $$T - 9$$
$31$ $$T + 3$$
$37$ $$T - 8$$
$41$ $$T - 3$$
$43$ $$T + 8$$
$47$ $$T - 7$$
$53$ $$T - 2$$
$59$ $$T + 4$$
$61$ $$T + 10$$
$67$ $$T - 8$$
$71$ $$T + 7$$
$73$ $$T + 9$$
$79$ $$T - 6$$
$83$ $$T + 14$$
$89$ $$T - 16$$
$97$ $$T + 6$$