# Properties

 Label 9450.2.a.dc Level $9450$ Weight $2$ Character orbit 9450.a Self dual yes Analytic conductor $75.459$ 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] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, 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("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9450.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: $$75.4586299101$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^7 + q^8 $$q + q^{2} + q^{4} + q^{7} + q^{8} - 5 q^{11} + q^{14} + q^{16} + 2 q^{17} - q^{19} - 5 q^{22} - q^{23} + q^{28} - 4 q^{29} - 9 q^{31} + q^{32} + 2 q^{34} - 5 q^{37} - q^{38} + 9 q^{41} + 10 q^{43} - 5 q^{44} - q^{46} + 6 q^{47} + q^{49} + 12 q^{53} + q^{56} - 4 q^{58} + 14 q^{59} - 9 q^{62} + q^{64} + 8 q^{67} + 2 q^{68} + 13 q^{71} + 2 q^{73} - 5 q^{74} - q^{76} - 5 q^{77} + 6 q^{79} + 9 q^{82} - 4 q^{83} + 10 q^{86} - 5 q^{88} + 9 q^{89} - q^{92} + 6 q^{94} - 16 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^7 + q^8 - 5 * q^11 + q^14 + q^16 + 2 * q^17 - q^19 - 5 * q^22 - q^23 + q^28 - 4 * q^29 - 9 * q^31 + q^32 + 2 * q^34 - 5 * q^37 - q^38 + 9 * q^41 + 10 * q^43 - 5 * q^44 - q^46 + 6 * q^47 + q^49 + 12 * q^53 + q^56 - 4 * q^58 + 14 * q^59 - 9 * q^62 + q^64 + 8 * q^67 + 2 * q^68 + 13 * q^71 + 2 * q^73 - 5 * q^74 - q^76 - 5 * q^77 + 6 * q^79 + 9 * q^82 - 4 * q^83 + 10 * q^86 - 5 * q^88 + 9 * q^89 - q^92 + 6 * q^94 - 16 * q^97 + 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 0 0 1.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 9450.2.a.dc 1
3.b odd 2 1 9450.2.a.bx 1
5.b even 2 1 378.2.a.c 1
15.d odd 2 1 378.2.a.f yes 1
20.d odd 2 1 3024.2.a.m 1
35.c odd 2 1 2646.2.a.i 1
45.h odd 6 2 1134.2.f.c 2
45.j even 6 2 1134.2.f.n 2
60.h even 2 1 3024.2.a.t 1
105.g even 2 1 2646.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.c 1 5.b even 2 1
378.2.a.f yes 1 15.d odd 2 1
1134.2.f.c 2 45.h odd 6 2
1134.2.f.n 2 45.j even 6 2
2646.2.a.i 1 35.c odd 2 1
2646.2.a.v 1 105.g even 2 1
3024.2.a.m 1 20.d odd 2 1
3024.2.a.t 1 60.h even 2 1
9450.2.a.bx 1 3.b odd 2 1
9450.2.a.dc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9450))$$:

 $$T_{11} + 5$$ T11 + 5 $$T_{13}$$ T13 $$T_{17} - 2$$ T17 - 2 $$T_{19} + 1$$ T19 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 5$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T + 1$$
$23$ $$T + 1$$
$29$ $$T + 4$$
$31$ $$T + 9$$
$37$ $$T + 5$$
$41$ $$T - 9$$
$43$ $$T - 10$$
$47$ $$T - 6$$
$53$ $$T - 12$$
$59$ $$T - 14$$
$61$ $$T$$
$67$ $$T - 8$$
$71$ $$T - 13$$
$73$ $$T - 2$$
$79$ $$T - 6$$
$83$ $$T + 4$$
$89$ $$T - 9$$
$97$ $$T + 16$$