# Properties

 Label 9200.2.a.p Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 46) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{7} - 3 q^{9}+O(q^{10})$$ q - 4 * q^7 - 3 * q^9 $$q - 4 q^{7} - 3 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{17} + 2 q^{19} + q^{23} + 2 q^{29} + 4 q^{37} + 6 q^{41} + 10 q^{43} + 9 q^{49} + 4 q^{53} - 12 q^{59} - 8 q^{61} + 12 q^{63} - 10 q^{67} - 6 q^{73} + 8 q^{77} + 12 q^{79} + 9 q^{81} + 14 q^{83} - 6 q^{89} - 8 q^{91} - 6 q^{97} + 6 q^{99}+O(q^{100})$$ q - 4 * q^7 - 3 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^17 + 2 * q^19 + q^23 + 2 * q^29 + 4 * q^37 + 6 * q^41 + 10 * q^43 + 9 * q^49 + 4 * q^53 - 12 * q^59 - 8 * q^61 + 12 * q^63 - 10 * q^67 - 6 * q^73 + 8 * q^77 + 12 * q^79 + 9 * q^81 + 14 * q^83 - 6 * q^89 - 8 * q^91 - 6 * q^97 + 6 * 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
0 0 0 0 0 −4.00000 0 −3.00000 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.p 1
4.b odd 2 1 1150.2.a.h 1
5.b even 2 1 368.2.a.d 1
15.d odd 2 1 3312.2.a.b 1
20.d odd 2 1 46.2.a.a 1
20.e even 4 2 1150.2.b.d 2
40.e odd 2 1 1472.2.a.f 1
40.f even 2 1 1472.2.a.g 1
60.h even 2 1 414.2.a.b 1
115.c odd 2 1 8464.2.a.g 1
140.c even 2 1 2254.2.a.c 1
220.g even 2 1 5566.2.a.h 1
260.g odd 2 1 7774.2.a.d 1
460.g even 2 1 1058.2.a.c 1
1380.b odd 2 1 9522.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.a.a 1 20.d odd 2 1
368.2.a.d 1 5.b even 2 1
414.2.a.b 1 60.h even 2 1
1058.2.a.c 1 460.g even 2 1
1150.2.a.h 1 4.b odd 2 1
1150.2.b.d 2 20.e even 4 2
1472.2.a.f 1 40.e odd 2 1
1472.2.a.g 1 40.f even 2 1
2254.2.a.c 1 140.c even 2 1
3312.2.a.b 1 15.d odd 2 1
5566.2.a.h 1 220.g even 2 1
7774.2.a.d 1 260.g odd 2 1
8464.2.a.g 1 115.c odd 2 1
9200.2.a.p 1 1.a even 1 1 trivial
9522.2.a.p 1 1380.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}$$ T3 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

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