# Properties

 Label 1225.2.a.b Level $1225$ Weight $2$ Character orbit 1225.a Self dual yes Analytic conductor $9.782$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} - 2 q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 + q^6 + 3 * q^8 - 2 * q^9 $$q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} - 2 q^{9} + q^{12} + 2 q^{13} - q^{16} + 2 q^{17} + 2 q^{18} - 6 q^{19} + 3 q^{23} - 3 q^{24} - 2 q^{26} + 5 q^{27} + 7 q^{29} - 2 q^{31} - 5 q^{32} - 2 q^{34} + 2 q^{36} + 8 q^{37} + 6 q^{38} - 2 q^{39} - 5 q^{41} - 7 q^{43} - 3 q^{46} + q^{48} - 2 q^{51} - 2 q^{52} - 6 q^{53} - 5 q^{54} + 6 q^{57} - 7 q^{58} - 10 q^{59} - 7 q^{61} + 2 q^{62} + 7 q^{64} + 5 q^{67} - 2 q^{68} - 3 q^{69} - 2 q^{71} - 6 q^{72} - 6 q^{73} - 8 q^{74} + 6 q^{76} + 2 q^{78} - 2 q^{79} + q^{81} + 5 q^{82} - 11 q^{83} + 7 q^{86} - 7 q^{87} - 9 q^{89} - 3 q^{92} + 2 q^{93} + 5 q^{96} - 16 q^{97}+O(q^{100})$$ q - q^2 - q^3 - q^4 + q^6 + 3 * q^8 - 2 * q^9 + q^12 + 2 * q^13 - q^16 + 2 * q^17 + 2 * q^18 - 6 * q^19 + 3 * q^23 - 3 * q^24 - 2 * q^26 + 5 * q^27 + 7 * q^29 - 2 * q^31 - 5 * q^32 - 2 * q^34 + 2 * q^36 + 8 * q^37 + 6 * q^38 - 2 * q^39 - 5 * q^41 - 7 * q^43 - 3 * q^46 + q^48 - 2 * q^51 - 2 * q^52 - 6 * q^53 - 5 * q^54 + 6 * q^57 - 7 * q^58 - 10 * q^59 - 7 * q^61 + 2 * q^62 + 7 * q^64 + 5 * q^67 - 2 * q^68 - 3 * q^69 - 2 * q^71 - 6 * q^72 - 6 * q^73 - 8 * q^74 + 6 * q^76 + 2 * q^78 - 2 * q^79 + q^81 + 5 * q^82 - 11 * q^83 + 7 * q^86 - 7 * q^87 - 9 * q^89 - 3 * q^92 + 2 * q^93 + 5 * q^96 - 16 * q^97

## 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 −1.00000 −1.00000 0 1.00000 0 3.00000 −2.00000 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.b 1
5.b even 2 1 1225.2.a.g 1
5.c odd 4 2 245.2.b.b 2
7.b odd 2 1 1225.2.a.d 1
7.d odd 6 2 175.2.e.b 2
15.e even 4 2 2205.2.d.e 2
35.c odd 2 1 1225.2.a.f 1
35.f even 4 2 245.2.b.c 2
35.i odd 6 2 175.2.e.a 2
35.k even 12 4 35.2.j.a 4
35.l odd 12 4 245.2.j.c 4
105.k odd 4 2 2205.2.d.d 2
105.w odd 12 4 315.2.bf.a 4
140.x odd 12 4 560.2.bw.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.j.a 4 35.k even 12 4
175.2.e.a 2 35.i odd 6 2
175.2.e.b 2 7.d odd 6 2
245.2.b.b 2 5.c odd 4 2
245.2.b.c 2 35.f even 4 2
245.2.j.c 4 35.l odd 12 4
315.2.bf.a 4 105.w odd 12 4
560.2.bw.b 4 140.x odd 12 4
1225.2.a.b 1 1.a even 1 1 trivial
1225.2.a.d 1 7.b odd 2 1
1225.2.a.f 1 35.c odd 2 1
1225.2.a.g 1 5.b even 2 1
2205.2.d.d 2 105.k odd 4 2
2205.2.d.e 2 15.e even 4 2

## Hecke kernels

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

 $$T_{2} + 1$$ T2 + 1 $$T_{3} + 1$$ T3 + 1

## Hecke characteristic polynomials

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