# Properties

 Label 1225.2.a.p Level $1225$ Weight $2$ Character orbit 1225.a Self dual yes Analytic conductor $9.782$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 1) q^{3} + ( - \beta + 2) q^{6} - 2 \beta q^{8} - 2 \beta q^{9} +O(q^{10})$$ q + b * q^2 + (b - 1) * q^3 + (-b + 2) * q^6 - 2*b * q^8 - 2*b * q^9 $$q + \beta q^{2} + (\beta - 1) q^{3} + ( - \beta + 2) q^{6} - 2 \beta q^{8} - 2 \beta q^{9} + (2 \beta - 3) q^{11} + ( - \beta - 3) q^{13} - 4 q^{16} + ( - 3 \beta + 1) q^{17} - 4 q^{18} + 6 q^{19} + ( - 3 \beta + 4) q^{22} + ( - \beta - 6) q^{23} + (2 \beta - 4) q^{24} + ( - 3 \beta - 2) q^{26} + ( - \beta - 1) q^{27} + ( - 4 \beta - 3) q^{29} + (3 \beta + 6) q^{31} + ( - 5 \beta + 7) q^{33} + (\beta - 6) q^{34} + ( - 3 \beta + 2) q^{37} + 6 \beta q^{38} + ( - 2 \beta + 1) q^{39} + ( - 3 \beta - 2) q^{41} - 2 q^{43} + ( - 6 \beta - 2) q^{46} + (3 \beta + 3) q^{47} + ( - 4 \beta + 4) q^{48} + (4 \beta - 7) q^{51} - 3 \beta q^{53} + ( - \beta - 2) q^{54} + (6 \beta - 6) q^{57} + ( - 3 \beta - 8) q^{58} + ( - 3 \beta + 2) q^{59} - 2 \beta q^{61} + (6 \beta + 6) q^{62} + 8 q^{64} + (7 \beta - 10) q^{66} + (3 \beta + 4) q^{67} + ( - 5 \beta + 4) q^{69} + (2 \beta - 6) q^{71} + 8 q^{72} + 6 \beta q^{73} + (2 \beta - 6) q^{74} + (\beta - 4) q^{78} + (6 \beta - 7) q^{79} + (6 \beta - 1) q^{81} + ( - 2 \beta - 6) q^{82} - 2 \beta q^{86} + (\beta - 5) q^{87} + (6 \beta - 8) q^{88} - 8 q^{89} + 3 \beta q^{93} + (3 \beta + 6) q^{94} + ( - 3 \beta - 9) q^{97} + (6 \beta - 8) q^{99} +O(q^{100})$$ q + b * q^2 + (b - 1) * q^3 + (-b + 2) * q^6 - 2*b * q^8 - 2*b * q^9 + (2*b - 3) * q^11 + (-b - 3) * q^13 - 4 * q^16 + (-3*b + 1) * q^17 - 4 * q^18 + 6 * q^19 + (-3*b + 4) * q^22 + (-b - 6) * q^23 + (2*b - 4) * q^24 + (-3*b - 2) * q^26 + (-b - 1) * q^27 + (-4*b - 3) * q^29 + (3*b + 6) * q^31 + (-5*b + 7) * q^33 + (b - 6) * q^34 + (-3*b + 2) * q^37 + 6*b * q^38 + (-2*b + 1) * q^39 + (-3*b - 2) * q^41 - 2 * q^43 + (-6*b - 2) * q^46 + (3*b + 3) * q^47 + (-4*b + 4) * q^48 + (4*b - 7) * q^51 - 3*b * q^53 + (-b - 2) * q^54 + (6*b - 6) * q^57 + (-3*b - 8) * q^58 + (-3*b + 2) * q^59 - 2*b * q^61 + (6*b + 6) * q^62 + 8 * q^64 + (7*b - 10) * q^66 + (3*b + 4) * q^67 + (-5*b + 4) * q^69 + (2*b - 6) * q^71 + 8 * q^72 + 6*b * q^73 + (2*b - 6) * q^74 + (b - 4) * q^78 + (6*b - 7) * q^79 + (6*b - 1) * q^81 + (-2*b - 6) * q^82 - 2*b * q^86 + (b - 5) * q^87 + (6*b - 8) * q^88 - 8 * q^89 + 3*b * q^93 + (3*b + 6) * q^94 + (-3*b - 9) * q^97 + (6*b - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{6}+O(q^{10})$$ 2 * q - 2 * q^3 + 4 * q^6 $$2 q - 2 q^{3} + 4 q^{6} - 6 q^{11} - 6 q^{13} - 8 q^{16} + 2 q^{17} - 8 q^{18} + 12 q^{19} + 8 q^{22} - 12 q^{23} - 8 q^{24} - 4 q^{26} - 2 q^{27} - 6 q^{29} + 12 q^{31} + 14 q^{33} - 12 q^{34} + 4 q^{37} + 2 q^{39} - 4 q^{41} - 4 q^{43} - 4 q^{46} + 6 q^{47} + 8 q^{48} - 14 q^{51} - 4 q^{54} - 12 q^{57} - 16 q^{58} + 4 q^{59} + 12 q^{62} + 16 q^{64} - 20 q^{66} + 8 q^{67} + 8 q^{69} - 12 q^{71} + 16 q^{72} - 12 q^{74} - 8 q^{78} - 14 q^{79} - 2 q^{81} - 12 q^{82} - 10 q^{87} - 16 q^{88} - 16 q^{89} + 12 q^{94} - 18 q^{97} - 16 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 4 * q^6 - 6 * q^11 - 6 * q^13 - 8 * q^16 + 2 * q^17 - 8 * q^18 + 12 * q^19 + 8 * q^22 - 12 * q^23 - 8 * q^24 - 4 * q^26 - 2 * q^27 - 6 * q^29 + 12 * q^31 + 14 * q^33 - 12 * q^34 + 4 * q^37 + 2 * q^39 - 4 * q^41 - 4 * q^43 - 4 * q^46 + 6 * q^47 + 8 * q^48 - 14 * q^51 - 4 * q^54 - 12 * q^57 - 16 * q^58 + 4 * q^59 + 12 * q^62 + 16 * q^64 - 20 * q^66 + 8 * q^67 + 8 * q^69 - 12 * q^71 + 16 * q^72 - 12 * q^74 - 8 * q^78 - 14 * q^79 - 2 * q^81 - 12 * q^82 - 10 * q^87 - 16 * q^88 - 16 * q^89 + 12 * q^94 - 18 * q^97 - 16 * 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
 −1.41421 1.41421
−1.41421 −2.41421 0 0 3.41421 0 2.82843 2.82843 0
1.2 1.41421 0.414214 0 0 0.585786 0 −2.82843 −2.82843 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.p 2
5.b even 2 1 245.2.a.f yes 2
5.c odd 4 2 1225.2.b.j 4
7.b odd 2 1 1225.2.a.r 2
15.d odd 2 1 2205.2.a.t 2
20.d odd 2 1 3920.2.a.br 2
35.c odd 2 1 245.2.a.e 2
35.f even 4 2 1225.2.b.i 4
35.i odd 6 2 245.2.e.g 4
35.j even 6 2 245.2.e.f 4
105.g even 2 1 2205.2.a.v 2
140.c even 2 1 3920.2.a.bw 2

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

## Hecke kernels

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

 $$T_{2}^{2} - 2$$ T2^2 - 2 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 6T + 1$$
$13$ $$T^{2} + 6T + 7$$
$17$ $$T^{2} - 2T - 17$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 12T + 34$$
$29$ $$T^{2} + 6T - 23$$
$31$ $$T^{2} - 12T + 18$$
$37$ $$T^{2} - 4T - 14$$
$41$ $$T^{2} + 4T - 14$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} - 6T - 9$$
$53$ $$T^{2} - 18$$
$59$ $$T^{2} - 4T - 14$$
$61$ $$T^{2} - 8$$
$67$ $$T^{2} - 8T - 2$$
$71$ $$T^{2} + 12T + 28$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} + 14T - 23$$
$83$ $$T^{2}$$
$89$ $$(T + 8)^{2}$$
$97$ $$T^{2} + 18T + 63$$