# Properties

 Label 1225.2.a.s 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: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 1) q^{3} + (\beta + 2) q^{4} + 4 q^{6} + (\beta + 4) q^{8} + ( - \beta + 2) q^{9}+O(q^{10})$$ q + b * q^2 + (b - 1) * q^3 + (b + 2) * q^4 + 4 * q^6 + (b + 4) * q^8 + (-b + 2) * q^9 $$q + \beta q^{2} + (\beta - 1) q^{3} + (\beta + 2) q^{4} + 4 q^{6} + (\beta + 4) q^{8} + ( - \beta + 2) q^{9} + ( - \beta + 1) q^{11} + (2 \beta + 2) q^{12} + ( - \beta + 3) q^{13} + 3 \beta q^{16} + (\beta - 3) q^{17} + (\beta - 4) q^{18} + (2 \beta + 2) q^{19} - 4 q^{22} + ( - 2 \beta + 2) q^{23} + 4 \beta q^{24} + (2 \beta - 4) q^{26} + ( - \beta - 3) q^{27} + (3 \beta - 1) q^{29} + (\beta + 4) q^{32} + (\beta - 5) q^{33} + ( - 2 \beta + 4) q^{34} - \beta q^{36} - 6 q^{37} + (4 \beta + 8) q^{38} + (3 \beta - 7) q^{39} - 2 \beta q^{41} + (2 \beta - 6) q^{43} + ( - 2 \beta - 2) q^{44} - 8 q^{46} + ( - 3 \beta - 1) q^{47} + 12 q^{48} + ( - 3 \beta + 7) q^{51} + 2 q^{52} + 2 \beta q^{53} + ( - 4 \beta - 4) q^{54} + (2 \beta + 6) q^{57} + (2 \beta + 12) q^{58} + 4 q^{59} - 6 \beta q^{61} + ( - \beta + 4) q^{64} + ( - 4 \beta + 4) q^{66} - 4 \beta q^{67} - 2 q^{68} + (2 \beta - 10) q^{69} + 8 q^{71} + ( - 3 \beta + 4) q^{72} + ( - 4 \beta - 2) q^{73} - 6 \beta q^{74} + (8 \beta + 12) q^{76} + ( - 4 \beta + 12) q^{78} + (\beta - 5) q^{79} - 7 q^{81} + ( - 2 \beta - 8) q^{82} + 4 q^{83} + ( - 4 \beta + 8) q^{86} + ( - \beta + 13) q^{87} - 4 \beta q^{88} + (2 \beta - 4) q^{89} + ( - 4 \beta - 4) q^{92} + ( - 4 \beta - 12) q^{94} + 4 \beta q^{96} + (5 \beta - 7) q^{97} + ( - 2 \beta + 6) q^{99} +O(q^{100})$$ q + b * q^2 + (b - 1) * q^3 + (b + 2) * q^4 + 4 * q^6 + (b + 4) * q^8 + (-b + 2) * q^9 + (-b + 1) * q^11 + (2*b + 2) * q^12 + (-b + 3) * q^13 + 3*b * q^16 + (b - 3) * q^17 + (b - 4) * q^18 + (2*b + 2) * q^19 - 4 * q^22 + (-2*b + 2) * q^23 + 4*b * q^24 + (2*b - 4) * q^26 + (-b - 3) * q^27 + (3*b - 1) * q^29 + (b + 4) * q^32 + (b - 5) * q^33 + (-2*b + 4) * q^34 - b * q^36 - 6 * q^37 + (4*b + 8) * q^38 + (3*b - 7) * q^39 - 2*b * q^41 + (2*b - 6) * q^43 + (-2*b - 2) * q^44 - 8 * q^46 + (-3*b - 1) * q^47 + 12 * q^48 + (-3*b + 7) * q^51 + 2 * q^52 + 2*b * q^53 + (-4*b - 4) * q^54 + (2*b + 6) * q^57 + (2*b + 12) * q^58 + 4 * q^59 - 6*b * q^61 + (-b + 4) * q^64 + (-4*b + 4) * q^66 - 4*b * q^67 - 2 * q^68 + (2*b - 10) * q^69 + 8 * q^71 + (-3*b + 4) * q^72 + (-4*b - 2) * q^73 - 6*b * q^74 + (8*b + 12) * q^76 + (-4*b + 12) * q^78 + (b - 5) * q^79 - 7 * q^81 + (-2*b - 8) * q^82 + 4 * q^83 + (-4*b + 8) * q^86 + (-b + 13) * q^87 - 4*b * q^88 + (2*b - 4) * q^89 + (-4*b - 4) * q^92 + (-4*b - 12) * q^94 + 4*b * q^96 + (5*b - 7) * q^97 + (-2*b + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{3} + 5 q^{4} + 8 q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + q^2 - q^3 + 5 * q^4 + 8 * q^6 + 9 * q^8 + 3 * q^9 $$2 q + q^{2} - q^{3} + 5 q^{4} + 8 q^{6} + 9 q^{8} + 3 q^{9} + q^{11} + 6 q^{12} + 5 q^{13} + 3 q^{16} - 5 q^{17} - 7 q^{18} + 6 q^{19} - 8 q^{22} + 2 q^{23} + 4 q^{24} - 6 q^{26} - 7 q^{27} + q^{29} + 9 q^{32} - 9 q^{33} + 6 q^{34} - q^{36} - 12 q^{37} + 20 q^{38} - 11 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{44} - 16 q^{46} - 5 q^{47} + 24 q^{48} + 11 q^{51} + 4 q^{52} + 2 q^{53} - 12 q^{54} + 14 q^{57} + 26 q^{58} + 8 q^{59} - 6 q^{61} + 7 q^{64} + 4 q^{66} - 4 q^{67} - 4 q^{68} - 18 q^{69} + 16 q^{71} + 5 q^{72} - 8 q^{73} - 6 q^{74} + 32 q^{76} + 20 q^{78} - 9 q^{79} - 14 q^{81} - 18 q^{82} + 8 q^{83} + 12 q^{86} + 25 q^{87} - 4 q^{88} - 6 q^{89} - 12 q^{92} - 28 q^{94} + 4 q^{96} - 9 q^{97} + 10 q^{99}+O(q^{100})$$ 2 * q + q^2 - q^3 + 5 * q^4 + 8 * q^6 + 9 * q^8 + 3 * q^9 + q^11 + 6 * q^12 + 5 * q^13 + 3 * q^16 - 5 * q^17 - 7 * q^18 + 6 * q^19 - 8 * q^22 + 2 * q^23 + 4 * q^24 - 6 * q^26 - 7 * q^27 + q^29 + 9 * q^32 - 9 * q^33 + 6 * q^34 - q^36 - 12 * q^37 + 20 * q^38 - 11 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^44 - 16 * q^46 - 5 * q^47 + 24 * q^48 + 11 * q^51 + 4 * q^52 + 2 * q^53 - 12 * q^54 + 14 * q^57 + 26 * q^58 + 8 * q^59 - 6 * q^61 + 7 * q^64 + 4 * q^66 - 4 * q^67 - 4 * q^68 - 18 * q^69 + 16 * q^71 + 5 * q^72 - 8 * q^73 - 6 * q^74 + 32 * q^76 + 20 * q^78 - 9 * q^79 - 14 * q^81 - 18 * q^82 + 8 * q^83 + 12 * q^86 + 25 * q^87 - 4 * q^88 - 6 * q^89 - 12 * q^92 - 28 * q^94 + 4 * q^96 - 9 * q^97 + 10 * 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
 −1.56155 2.56155
−1.56155 −2.56155 0.438447 0 4.00000 0 2.43845 3.56155 0
1.2 2.56155 1.56155 4.56155 0 4.00000 0 6.56155 −0.561553 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.s 2
5.b even 2 1 245.2.a.d 2
5.c odd 4 2 1225.2.b.f 4
7.b odd 2 1 175.2.a.f 2
15.d odd 2 1 2205.2.a.x 2
20.d odd 2 1 3920.2.a.bs 2
21.c even 2 1 1575.2.a.p 2
28.d even 2 1 2800.2.a.bi 2
35.c odd 2 1 35.2.a.b 2
35.f even 4 2 175.2.b.b 4
35.i odd 6 2 245.2.e.i 4
35.j even 6 2 245.2.e.h 4
105.g even 2 1 315.2.a.e 2
105.k odd 4 2 1575.2.d.e 4
140.c even 2 1 560.2.a.i 2
140.j odd 4 2 2800.2.g.t 4
280.c odd 2 1 2240.2.a.bh 2
280.n even 2 1 2240.2.a.bd 2
385.h even 2 1 4235.2.a.m 2
420.o odd 2 1 5040.2.a.bt 2
455.h odd 2 1 5915.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 35.c odd 2 1
175.2.a.f 2 7.b odd 2 1
175.2.b.b 4 35.f even 4 2
245.2.a.d 2 5.b even 2 1
245.2.e.h 4 35.j even 6 2
245.2.e.i 4 35.i odd 6 2
315.2.a.e 2 105.g even 2 1
560.2.a.i 2 140.c even 2 1
1225.2.a.s 2 1.a even 1 1 trivial
1225.2.b.f 4 5.c odd 4 2
1575.2.a.p 2 21.c even 2 1
1575.2.d.e 4 105.k odd 4 2
2205.2.a.x 2 15.d odd 2 1
2240.2.a.bd 2 280.n even 2 1
2240.2.a.bh 2 280.c odd 2 1
2800.2.a.bi 2 28.d even 2 1
2800.2.g.t 4 140.j odd 4 2
3920.2.a.bs 2 20.d odd 2 1
4235.2.a.m 2 385.h even 2 1
5040.2.a.bt 2 420.o odd 2 1
5915.2.a.l 2 455.h odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 4$$ T2^2 - T2 - 4 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$T^{2} + T - 4$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T - 4$$
$13$ $$T^{2} - 5T + 2$$
$17$ $$T^{2} + 5T + 2$$
$19$ $$T^{2} - 6T - 8$$
$23$ $$T^{2} - 2T - 16$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2}$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 2T - 16$$
$43$ $$T^{2} + 10T + 8$$
$47$ $$T^{2} + 5T - 32$$
$53$ $$T^{2} - 2T - 16$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} + 6T - 144$$
$67$ $$T^{2} + 4T - 64$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 8T - 52$$
$79$ $$T^{2} + 9T + 16$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 6T - 8$$
$97$ $$T^{2} + 9T - 86$$