# Properties

 Label 9200.2.a.b 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 92) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 4 q^{7} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 - 4 * q^7 + 6 * q^9 $$q - 3 q^{3} - 4 q^{7} + 6 q^{9} - 2 q^{11} + 5 q^{13} - 4 q^{17} + 2 q^{19} + 12 q^{21} + q^{23} - 9 q^{27} - 7 q^{29} + 3 q^{31} + 6 q^{33} - 2 q^{37} - 15 q^{39} - 9 q^{41} - 8 q^{43} + 9 q^{47} + 9 q^{49} + 12 q^{51} - 2 q^{53} - 6 q^{57} - 2 q^{61} - 24 q^{63} + 14 q^{67} - 3 q^{69} + 3 q^{71} + 3 q^{73} + 8 q^{77} + 6 q^{79} + 9 q^{81} + 8 q^{83} + 21 q^{87} + 12 q^{89} - 20 q^{91} - 9 q^{93} - 12 q^{99}+O(q^{100})$$ q - 3 * q^3 - 4 * q^7 + 6 * q^9 - 2 * q^11 + 5 * q^13 - 4 * q^17 + 2 * q^19 + 12 * q^21 + q^23 - 9 * q^27 - 7 * q^29 + 3 * q^31 + 6 * q^33 - 2 * q^37 - 15 * q^39 - 9 * q^41 - 8 * q^43 + 9 * q^47 + 9 * q^49 + 12 * q^51 - 2 * q^53 - 6 * q^57 - 2 * q^61 - 24 * q^63 + 14 * q^67 - 3 * q^69 + 3 * q^71 + 3 * q^73 + 8 * q^77 + 6 * q^79 + 9 * q^81 + 8 * q^83 + 21 * q^87 + 12 * q^89 - 20 * q^91 - 9 * q^93 - 12 * 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 −3.00000 0 0 0 −4.00000 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.b 1
4.b odd 2 1 2300.2.a.h 1
5.b even 2 1 368.2.a.g 1
15.d odd 2 1 3312.2.a.q 1
20.d odd 2 1 92.2.a.a 1
20.e even 4 2 2300.2.c.b 2
40.e odd 2 1 1472.2.a.n 1
40.f even 2 1 1472.2.a.b 1
60.h even 2 1 828.2.a.c 1
115.c odd 2 1 8464.2.a.s 1
140.c even 2 1 4508.2.a.d 1
460.g even 2 1 2116.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.a.a 1 20.d odd 2 1
368.2.a.g 1 5.b even 2 1
828.2.a.c 1 60.h even 2 1
1472.2.a.b 1 40.f even 2 1
1472.2.a.n 1 40.e odd 2 1
2116.2.a.a 1 460.g even 2 1
2300.2.a.h 1 4.b odd 2 1
2300.2.c.b 2 20.e even 4 2
3312.2.a.q 1 15.d odd 2 1
4508.2.a.d 1 140.c even 2 1
8464.2.a.s 1 115.c odd 2 1
9200.2.a.b 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 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T + 2$$
$13$ $$T - 5$$
$17$ $$T + 4$$
$19$ $$T - 2$$
$23$ $$T - 1$$
$29$ $$T + 7$$
$31$ $$T - 3$$
$37$ $$T + 2$$
$41$ $$T + 9$$
$43$ $$T + 8$$
$47$ $$T - 9$$
$53$ $$T + 2$$
$59$ $$T$$
$61$ $$T + 2$$
$67$ $$T - 14$$
$71$ $$T - 3$$
$73$ $$T - 3$$
$79$ $$T - 6$$
$83$ $$T - 8$$
$89$ $$T - 12$$
$97$ $$T$$