# Properties

 Label 225.2.a.a Level $225$ Weight $2$ Character orbit 225.a Self dual yes Analytic conductor $1.797$ 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] = [225,2,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.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: $$1.79663404548$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} - 3 q^{7}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 - 3 * q^7 $$q - 2 q^{2} + 2 q^{4} - 3 q^{7} - 2 q^{11} + q^{13} + 6 q^{14} - 4 q^{16} - 2 q^{17} - 5 q^{19} + 4 q^{22} - 6 q^{23} - 2 q^{26} - 6 q^{28} - 10 q^{29} - 3 q^{31} + 8 q^{32} + 4 q^{34} + 2 q^{37} + 10 q^{38} + 8 q^{41} + q^{43} - 4 q^{44} + 12 q^{46} - 2 q^{47} + 2 q^{49} + 2 q^{52} + 4 q^{53} + 20 q^{58} + 10 q^{59} + 7 q^{61} + 6 q^{62} - 8 q^{64} - 3 q^{67} - 4 q^{68} + 8 q^{71} - 14 q^{73} - 4 q^{74} - 10 q^{76} + 6 q^{77} - 16 q^{82} - 6 q^{83} - 2 q^{86} - 3 q^{91} - 12 q^{92} + 4 q^{94} + 17 q^{97} - 4 q^{98}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 - 3 * q^7 - 2 * q^11 + q^13 + 6 * q^14 - 4 * q^16 - 2 * q^17 - 5 * q^19 + 4 * q^22 - 6 * q^23 - 2 * q^26 - 6 * q^28 - 10 * q^29 - 3 * q^31 + 8 * q^32 + 4 * q^34 + 2 * q^37 + 10 * q^38 + 8 * q^41 + q^43 - 4 * q^44 + 12 * q^46 - 2 * q^47 + 2 * q^49 + 2 * q^52 + 4 * q^53 + 20 * q^58 + 10 * q^59 + 7 * q^61 + 6 * q^62 - 8 * q^64 - 3 * q^67 - 4 * q^68 + 8 * q^71 - 14 * q^73 - 4 * q^74 - 10 * q^76 + 6 * q^77 - 16 * q^82 - 6 * q^83 - 2 * q^86 - 3 * q^91 - 12 * q^92 + 4 * q^94 + 17 * q^97 - 4 * 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
−2.00000 0 2.00000 0 0 −3.00000 0 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
$$3$$ $$-1$$
$$5$$ $$-1$$

## 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 225.2.a.a 1
3.b odd 2 1 75.2.a.c yes 1
4.b odd 2 1 3600.2.a.bk 1
5.b even 2 1 225.2.a.e 1
5.c odd 4 2 225.2.b.a 2
12.b even 2 1 1200.2.a.p 1
15.d odd 2 1 75.2.a.a 1
15.e even 4 2 75.2.b.a 2
20.d odd 2 1 3600.2.a.j 1
20.e even 4 2 3600.2.f.p 2
21.c even 2 1 3675.2.a.q 1
24.f even 2 1 4800.2.a.be 1
24.h odd 2 1 4800.2.a.bq 1
33.d even 2 1 9075.2.a.a 1
60.h even 2 1 1200.2.a.c 1
60.l odd 4 2 1200.2.f.d 2
105.g even 2 1 3675.2.a.b 1
120.i odd 2 1 4800.2.a.bb 1
120.m even 2 1 4800.2.a.br 1
120.q odd 4 2 4800.2.f.y 2
120.w even 4 2 4800.2.f.l 2
165.d even 2 1 9075.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 15.d odd 2 1
75.2.a.c yes 1 3.b odd 2 1
75.2.b.a 2 15.e even 4 2
225.2.a.a 1 1.a even 1 1 trivial
225.2.a.e 1 5.b even 2 1
225.2.b.a 2 5.c odd 4 2
1200.2.a.c 1 60.h even 2 1
1200.2.a.p 1 12.b even 2 1
1200.2.f.d 2 60.l odd 4 2
3600.2.a.j 1 20.d odd 2 1
3600.2.a.bk 1 4.b odd 2 1
3600.2.f.p 2 20.e even 4 2
3675.2.a.b 1 105.g even 2 1
3675.2.a.q 1 21.c even 2 1
4800.2.a.bb 1 120.i odd 2 1
4800.2.a.be 1 24.f even 2 1
4800.2.a.bq 1 24.h odd 2 1
4800.2.a.br 1 120.m even 2 1
4800.2.f.l 2 120.w even 4 2
4800.2.f.y 2 120.q odd 4 2
9075.2.a.a 1 33.d even 2 1
9075.2.a.s 1 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7} + 3$$ T7 + 3

## Hecke characteristic polynomials

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