# Properties

 Label 1225.2.a.j Level $1225$ Weight $2$ Character orbit 1225.a Self dual yes Analytic conductor $9.782$ 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] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.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: $$9.78167424761$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 6 q^{6} + 6 q^{9}+O(q^{10})$$ q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 6 * q^6 + 6 * q^9 $$q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 6 q^{6} + 6 q^{9} + q^{11} + 6 q^{12} + 3 q^{13} - 4 q^{16} - 3 q^{17} + 12 q^{18} - 6 q^{19} + 2 q^{22} + 4 q^{23} + 6 q^{26} + 9 q^{27} - q^{29} - 6 q^{31} - 8 q^{32} + 3 q^{33} - 6 q^{34} + 12 q^{36} - 12 q^{38} + 9 q^{39} - 6 q^{41} + 6 q^{43} + 2 q^{44} + 8 q^{46} - 9 q^{47} - 12 q^{48} - 9 q^{51} + 6 q^{52} + 10 q^{53} + 18 q^{54} - 18 q^{57} - 2 q^{58} + 6 q^{59} - 12 q^{62} - 8 q^{64} + 6 q^{66} + 14 q^{67} - 6 q^{68} + 12 q^{69} - 8 q^{71} + 6 q^{73} - 12 q^{76} + 18 q^{78} - q^{79} + 9 q^{81} - 12 q^{82} + 12 q^{83} + 12 q^{86} - 3 q^{87} - 12 q^{89} + 8 q^{92} - 18 q^{93} - 18 q^{94} - 24 q^{96} - 15 q^{97} + 6 q^{99}+O(q^{100})$$ q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 6 * q^6 + 6 * q^9 + q^11 + 6 * q^12 + 3 * q^13 - 4 * q^16 - 3 * q^17 + 12 * q^18 - 6 * q^19 + 2 * q^22 + 4 * q^23 + 6 * q^26 + 9 * q^27 - q^29 - 6 * q^31 - 8 * q^32 + 3 * q^33 - 6 * q^34 + 12 * q^36 - 12 * q^38 + 9 * q^39 - 6 * q^41 + 6 * q^43 + 2 * q^44 + 8 * q^46 - 9 * q^47 - 12 * q^48 - 9 * q^51 + 6 * q^52 + 10 * q^53 + 18 * q^54 - 18 * q^57 - 2 * q^58 + 6 * q^59 - 12 * q^62 - 8 * q^64 + 6 * q^66 + 14 * q^67 - 6 * q^68 + 12 * q^69 - 8 * q^71 + 6 * q^73 - 12 * q^76 + 18 * q^78 - q^79 + 9 * q^81 - 12 * q^82 + 12 * q^83 + 12 * q^86 - 3 * q^87 - 12 * q^89 + 8 * q^92 - 18 * q^93 - 18 * q^94 - 24 * q^96 - 15 * 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
2.00000 3.00000 2.00000 0 6.00000 0 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
$$5$$ $$+1$$
$$7$$ $$-1$$

## 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 1225.2.a.j 1
5.b even 2 1 245.2.a.a 1
5.c odd 4 2 1225.2.b.b 2
7.b odd 2 1 1225.2.a.h 1
15.d odd 2 1 2205.2.a.j 1
20.d odd 2 1 3920.2.a.bj 1
35.c odd 2 1 245.2.a.b yes 1
35.f even 4 2 1225.2.b.a 2
35.i odd 6 2 245.2.e.c 2
35.j even 6 2 245.2.e.d 2
105.g even 2 1 2205.2.a.l 1
140.c even 2 1 3920.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 5.b even 2 1
245.2.a.b yes 1 35.c odd 2 1
245.2.e.c 2 35.i odd 6 2
245.2.e.d 2 35.j even 6 2
1225.2.a.h 1 7.b odd 2 1
1225.2.a.j 1 1.a even 1 1 trivial
1225.2.b.a 2 35.f even 4 2
1225.2.b.b 2 5.c odd 4 2
2205.2.a.j 1 15.d odd 2 1
2205.2.a.l 1 105.g even 2 1
3920.2.a.a 1 140.c even 2 1
3920.2.a.bj 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{3} - 3$$ T3 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T - 3$$
$17$ $$T + 3$$
$19$ $$T + 6$$
$23$ $$T - 4$$
$29$ $$T + 1$$
$31$ $$T + 6$$
$37$ $$T$$
$41$ $$T + 6$$
$43$ $$T - 6$$
$47$ $$T + 9$$
$53$ $$T - 10$$
$59$ $$T - 6$$
$61$ $$T$$
$67$ $$T - 14$$
$71$ $$T + 8$$
$73$ $$T - 6$$
$79$ $$T + 1$$
$83$ $$T - 12$$
$89$ $$T + 12$$
$97$ $$T + 15$$