# Properties

 Label 1225.2.a.m Level $1225$ Weight $2$ Character orbit 1225.a Self dual yes Analytic conductor $9.782$ Analytic rank $0$ 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: $$0$$ 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 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)

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

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + (\beta + 1) q^{3} + ( - 2 \beta + 1) q^{4} + q^{6} + (\beta - 3) q^{8} + 2 \beta q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + (b + 1) * q^3 + (-2*b + 1) * q^4 + q^6 + (b - 3) * q^8 + 2*b * q^9 $$q + (\beta - 1) q^{2} + (\beta + 1) q^{3} + ( - 2 \beta + 1) q^{4} + q^{6} + (\beta - 3) q^{8} + 2 \beta q^{9} + (2 \beta + 2) q^{11} + ( - \beta - 3) q^{12} + (2 \beta - 2) q^{13} + 3 q^{16} + ( - 2 \beta + 2) q^{17} + ( - 2 \beta + 4) q^{18} + 2 \beta q^{19} + 2 q^{22} + (\beta + 1) q^{23} + ( - 2 \beta - 1) q^{24} + ( - 4 \beta + 6) q^{26} + ( - \beta + 1) q^{27} - q^{29} + 6 q^{31} + (\beta + 3) q^{32} + (4 \beta + 6) q^{33} + (4 \beta - 6) q^{34} + (2 \beta - 8) q^{36} + ( - 2 \beta + 4) q^{38} + 2 q^{39} + ( - 2 \beta + 5) q^{41} + ( - \beta - 5) q^{43} + ( - 2 \beta - 6) q^{44} + q^{46} + 2 q^{47} + (3 \beta + 3) q^{48} - 2 q^{51} + (6 \beta - 10) q^{52} + (2 \beta + 4) q^{53} + (2 \beta - 3) q^{54} + (2 \beta + 4) q^{57} + ( - \beta + 1) q^{58} + (6 \beta + 4) q^{59} + (6 \beta + 3) q^{61} + (6 \beta - 6) q^{62} + (2 \beta - 7) q^{64} + (2 \beta + 2) q^{66} + ( - \beta - 11) q^{67} + ( - 6 \beta + 10) q^{68} + (2 \beta + 3) q^{69} + ( - 6 \beta - 4) q^{71} + ( - 6 \beta + 4) q^{72} + (2 \beta + 2) q^{73} + (2 \beta - 8) q^{76} + (2 \beta - 2) q^{78} + ( - 2 \beta + 12) q^{79} + ( - 6 \beta - 1) q^{81} + (7 \beta - 9) q^{82} + ( - 9 \beta + 1) q^{83} + ( - 4 \beta + 3) q^{86} + ( - \beta - 1) q^{87} + ( - 4 \beta - 2) q^{88} + ( - 4 \beta + 3) q^{89} + ( - \beta - 3) q^{92} + (6 \beta + 6) q^{93} + (2 \beta - 2) q^{94} + (4 \beta + 5) q^{96} + ( - 4 \beta + 6) q^{97} + (4 \beta + 8) q^{99}+O(q^{100})$$ q + (b - 1) * q^2 + (b + 1) * q^3 + (-2*b + 1) * q^4 + q^6 + (b - 3) * q^8 + 2*b * q^9 + (2*b + 2) * q^11 + (-b - 3) * q^12 + (2*b - 2) * q^13 + 3 * q^16 + (-2*b + 2) * q^17 + (-2*b + 4) * q^18 + 2*b * q^19 + 2 * q^22 + (b + 1) * q^23 + (-2*b - 1) * q^24 + (-4*b + 6) * q^26 + (-b + 1) * q^27 - q^29 + 6 * q^31 + (b + 3) * q^32 + (4*b + 6) * q^33 + (4*b - 6) * q^34 + (2*b - 8) * q^36 + (-2*b + 4) * q^38 + 2 * q^39 + (-2*b + 5) * q^41 + (-b - 5) * q^43 + (-2*b - 6) * q^44 + q^46 + 2 * q^47 + (3*b + 3) * q^48 - 2 * q^51 + (6*b - 10) * q^52 + (2*b + 4) * q^53 + (2*b - 3) * q^54 + (2*b + 4) * q^57 + (-b + 1) * q^58 + (6*b + 4) * q^59 + (6*b + 3) * q^61 + (6*b - 6) * q^62 + (2*b - 7) * q^64 + (2*b + 2) * q^66 + (-b - 11) * q^67 + (-6*b + 10) * q^68 + (2*b + 3) * q^69 + (-6*b - 4) * q^71 + (-6*b + 4) * q^72 + (2*b + 2) * q^73 + (2*b - 8) * q^76 + (2*b - 2) * q^78 + (-2*b + 12) * q^79 + (-6*b - 1) * q^81 + (7*b - 9) * q^82 + (-9*b + 1) * q^83 + (-4*b + 3) * q^86 + (-b - 1) * q^87 + (-4*b - 2) * q^88 + (-4*b + 3) * q^89 + (-b - 3) * q^92 + (6*b + 6) * q^93 + (2*b - 2) * q^94 + (4*b + 5) * q^96 + (-4*b + 6) * q^97 + (4*b + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{11} - 6 q^{12} - 4 q^{13} + 6 q^{16} + 4 q^{17} + 8 q^{18} + 4 q^{22} + 2 q^{23} - 2 q^{24} + 12 q^{26} + 2 q^{27} - 2 q^{29} + 12 q^{31} + 6 q^{32} + 12 q^{33} - 12 q^{34} - 16 q^{36} + 8 q^{38} + 4 q^{39} + 10 q^{41} - 10 q^{43} - 12 q^{44} + 2 q^{46} + 4 q^{47} + 6 q^{48} - 4 q^{51} - 20 q^{52} + 8 q^{53} - 6 q^{54} + 8 q^{57} + 2 q^{58} + 8 q^{59} + 6 q^{61} - 12 q^{62} - 14 q^{64} + 4 q^{66} - 22 q^{67} + 20 q^{68} + 6 q^{69} - 8 q^{71} + 8 q^{72} + 4 q^{73} - 16 q^{76} - 4 q^{78} + 24 q^{79} - 2 q^{81} - 18 q^{82} + 2 q^{83} + 6 q^{86} - 2 q^{87} - 4 q^{88} + 6 q^{89} - 6 q^{92} + 12 q^{93} - 4 q^{94} + 10 q^{96} + 12 q^{97} + 16 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^11 - 6 * q^12 - 4 * q^13 + 6 * q^16 + 4 * q^17 + 8 * q^18 + 4 * q^22 + 2 * q^23 - 2 * q^24 + 12 * q^26 + 2 * q^27 - 2 * q^29 + 12 * q^31 + 6 * q^32 + 12 * q^33 - 12 * q^34 - 16 * q^36 + 8 * q^38 + 4 * q^39 + 10 * q^41 - 10 * q^43 - 12 * q^44 + 2 * q^46 + 4 * q^47 + 6 * q^48 - 4 * q^51 - 20 * q^52 + 8 * q^53 - 6 * q^54 + 8 * q^57 + 2 * q^58 + 8 * q^59 + 6 * q^61 - 12 * q^62 - 14 * q^64 + 4 * q^66 - 22 * q^67 + 20 * q^68 + 6 * q^69 - 8 * q^71 + 8 * q^72 + 4 * q^73 - 16 * q^76 - 4 * q^78 + 24 * q^79 - 2 * q^81 - 18 * q^82 + 2 * q^83 + 6 * q^86 - 2 * q^87 - 4 * q^88 + 6 * q^89 - 6 * q^92 + 12 * q^93 - 4 * q^94 + 10 * q^96 + 12 * 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
−2.41421 −0.414214 3.82843 0 1.00000 0 −4.41421 −2.82843 0
1.2 0.414214 2.41421 −1.82843 0 1.00000 0 −1.58579 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.m 2
5.b even 2 1 245.2.a.g 2
5.c odd 4 2 1225.2.b.h 4
7.b odd 2 1 1225.2.a.k 2
7.d odd 6 2 175.2.e.c 4
15.d odd 2 1 2205.2.a.q 2
20.d odd 2 1 3920.2.a.bv 2
35.c odd 2 1 245.2.a.h 2
35.f even 4 2 1225.2.b.g 4
35.i odd 6 2 35.2.e.a 4
35.j even 6 2 245.2.e.e 4
35.k even 12 4 175.2.k.a 8
105.g even 2 1 2205.2.a.n 2
105.p even 6 2 315.2.j.e 4
140.c even 2 1 3920.2.a.bq 2
140.s even 6 2 560.2.q.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 35.i odd 6 2
175.2.e.c 4 7.d odd 6 2
175.2.k.a 8 35.k even 12 4
245.2.a.g 2 5.b even 2 1
245.2.a.h 2 35.c odd 2 1
245.2.e.e 4 35.j even 6 2
315.2.j.e 4 105.p even 6 2
560.2.q.k 4 140.s even 6 2
1225.2.a.k 2 7.b odd 2 1
1225.2.a.m 2 1.a even 1 1 trivial
1225.2.b.g 4 35.f even 4 2
1225.2.b.h 4 5.c odd 4 2
2205.2.a.n 2 105.g even 2 1
2205.2.a.q 2 15.d odd 2 1
3920.2.a.bq 2 140.c even 2 1
3920.2.a.bv 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1225))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$T^{2} - 2T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T - 4$$
$13$ $$T^{2} + 4T - 4$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$T^{2} - 8$$
$23$ $$T^{2} - 2T - 1$$
$29$ $$(T + 1)^{2}$$
$31$ $$(T - 6)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 10T + 17$$
$43$ $$T^{2} + 10T + 23$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} - 8T + 8$$
$59$ $$T^{2} - 8T - 56$$
$61$ $$T^{2} - 6T - 63$$
$67$ $$T^{2} + 22T + 119$$
$71$ $$T^{2} + 8T - 56$$
$73$ $$T^{2} - 4T - 4$$
$79$ $$T^{2} - 24T + 136$$
$83$ $$T^{2} - 2T - 161$$
$89$ $$T^{2} - 6T - 23$$
$97$ $$T^{2} - 12T + 4$$