# Properties

 Label 5225.2.a.j Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_{3} + \beta_1) q^{2} + (\beta_{4} + \beta_{2} - \beta_1 + 2) q^{3} + (\beta_{4} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} + \beta_1) q^{6} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 3) q^{7} + (\beta_{4} + 2 \beta_{2}) q^{8} + (\beta_{4} - \beta_{3} + 3 \beta_{2} + \cdots + 3) q^{9}+O(q^{10})$$ q + (-b4 + b3 + b1) * q^2 + (b4 + b2 - b1 + 2) * q^3 + (b4 + b1 + 1) * q^4 + (2*b3 + b2 + b1) * q^6 + (b4 - 2*b3 + b2 + 3) * q^7 + (b4 + 2*b2) * q^8 + (b4 - b3 + 3*b2 - 2*b1 + 3) * q^9 $$q + ( - \beta_{4} + \beta_{3} + \beta_1) q^{2} + (\beta_{4} + \beta_{2} - \beta_1 + 2) q^{3} + (\beta_{4} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} + \beta_1) q^{6} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 3) q^{7} + (\beta_{4} + 2 \beta_{2}) q^{8} + (\beta_{4} - \beta_{3} + 3 \beta_{2} + \cdots + 3) q^{9}+ \cdots + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + \cdots - 3) q^{99}+O(q^{100})$$ q + (-b4 + b3 + b1) * q^2 + (b4 + b2 - b1 + 2) * q^3 + (b4 + b1 + 1) * q^4 + (2*b3 + b2 + b1) * q^6 + (b4 - 2*b3 + b2 + 3) * q^7 + (b4 + 2*b2) * q^8 + (b4 - b3 + 3*b2 - 2*b1 + 3) * q^9 - q^11 + (2*b4 + b3 + 3*b1 + 1) * q^12 + (b3 - 3*b2 - 1) * q^13 + (-3*b4 + b3 + 2*b2 + 3*b1 - 1) * q^14 + (2*b4 - b3 + 3*b2 - 2*b1 + 1) * q^16 + (2*b4 + b3 + b2 + 1) * q^17 + (2*b4 + 2*b2 + b1 + 2) * q^18 + q^19 + (5*b4 - 2*b3 + 4*b2 - 3*b1 + 5) * q^21 + (b4 - b3 - b1) * q^22 + (-b4 + 2*b1 + 1) * q^23 + (3*b2 + 2*b1 + 2) * q^24 + (-4*b4 + 3*b3 - 3*b2 - b1 - 5) * q^26 + (-b4 - 3*b3 + 5*b2 - 3*b1 + 4) * q^27 + (4*b4 - b3 + 2*b1 + 5) * q^28 + (-b4 + 3*b3 - 2*b2 + 5*b1) * q^29 + (b3 + b2 - 5*b1) * q^31 + (2*b4 - b3 - b2 - b1 + 1) * q^32 + (-b4 - b2 + b1 - 2) * q^33 + (2*b4 + 3*b3 + 3*b2 + b1 + 1) * q^34 + (4*b3 - b2 + 7*b1 - 3) * q^36 + (-4*b3 - 2*b1 + 3) * q^37 + (-b4 + b3 + b1) * q^38 + (-2*b4 + b3 - 7*b2 + 2*b1 - 4) * q^39 + (3*b4 - b3 + 3*b2 - 3*b1 + 5) * q^41 + (b4 + 4*b3 + 6*b2 + 2*b1 - 2) * q^42 + (-6*b4 - 3*b2 - b1 + 1) * q^43 + (-b4 - b1 - 1) * q^44 + (-b4 + b2 + 3*b1 + 3) * q^46 + (2*b4 + b3 - 4*b1 + 5) * q^47 + (-3*b3 + 5*b2 - 2*b1 + 6) * q^48 + (b4 - 3*b3 - 2*b2 + 2*b1 + 4) * q^49 + (b3 + b2 + 4*b1 + 4) * q^51 + (2*b4 - 5*b3 - 2*b2 - 6*b1 + 2) * q^52 + (-6*b4 + 7*b3 - 3*b2 + 4*b1 - 3) * q^53 + (3*b4 - 5*b3 + b2 + b1 + 5) * q^54 + (6*b3 + 2*b2 + b1 - 1) * q^56 + (b4 + b2 - b1 + 2) * q^57 + (-b4 + 4*b3 + 2*b2 + 5*b1 + 5) * q^58 + (2*b4 + b3 - b2 - 2*b1 - 3) * q^59 + (-3*b3 + b1 + 1) * q^61 + (3*b4 - 4*b2 - 5*b1 - 2) * q^62 + (b4 + b3 + 5*b2 - 3*b1 + 6) * q^63 + (-8*b4 + 5*b3 - 6*b2 + 4*b1 - 8) * q^64 + (-2*b3 - b2 - b1) * q^66 + (-5*b3 - b2 - 3*b1 + 7) * q^67 + (4*b4 + b3 + 4*b2 + 2*b1 + 6) * q^68 + (3*b4 + 2*b3 + 2*b2 + b1) * q^69 + (2*b4 - b2 - 1) * q^71 + (b4 + 2*b3 + 2*b2 + 2*b1 + 5) * q^72 + (-2*b4 + b3 - 2*b2 + b1 + 2) * q^73 + (-7*b4 - b3 - 2*b2 + b1 - 6) * q^74 + (b4 + b1 + 1) * q^76 + (-b4 + 2*b3 - b2 - 3) * q^77 + (-9*b4 + 2*b3 - 7*b2 - 2*b1 - 9) * q^78 + (-2*b4 - 2*b3 + 4*b2 + 2*b1 + 1) * q^79 + (b4 - 3*b3 + 6*b2 - 9*b1 + 4) * q^81 + (4*b3 + 3*b2 + 2*b1 - 1) * q^82 + (5*b3 - b2 + 2*b1 + 5) * q^83 + (8*b4 + b3 + b2 + 6*b1 + 7) * q^84 + (-7*b4 - 2*b3 - 10*b2 - 1) * q^86 + (2*b4 + 8*b3 - 3*b2 + 11*b1 - 4) * q^87 + (-b4 - 2*b2) * q^88 + (-6*b4 - b3 - b2 - 2*b1 - 4) * q^89 + (7*b4 - 7*b3 - b2 - 4*b1 - 1) * q^91 + (b4 + b3 + 3*b2 + 2*b1 + 4) * q^92 + (-6*b4 - 4*b3 + 2*b2 - 9*b1 + 7) * q^93 + (-4*b4 + 8*b3 - 2*b2 + b1 - 5) * q^94 + (b4 - 2*b3 - 3*b2 + 1) * q^96 + (3*b4 - 5*b3 - 3*b1 + 5) * q^97 + (-11*b4 + 4*b3 + b2 + 6*b1 - 6) * q^98 + (-b4 + b3 - 3*b2 + 2*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10})$$ 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 $$5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26} + 10 q^{27} + 22 q^{28} + 11 q^{29} - 5 q^{31} + 2 q^{32} - 7 q^{33} + 4 q^{34} - 3 q^{36} + 9 q^{37} + 3 q^{38} - 8 q^{39} + 15 q^{41} - 11 q^{42} + 13 q^{43} - 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{48} + 20 q^{49} + 24 q^{51} - q^{52} + 5 q^{53} + 17 q^{54} + 7 q^{57} + 33 q^{58} - 17 q^{59} + 3 q^{61} - 14 q^{62} + 22 q^{63} - 17 q^{64} - 2 q^{66} + 28 q^{67} + 25 q^{68} - 2 q^{69} - 6 q^{71} + 26 q^{72} + 16 q^{73} - 21 q^{74} + 5 q^{76} - 11 q^{77} - 29 q^{78} + 3 q^{79} + q^{81} - 2 q^{82} + 33 q^{83} + 33 q^{84} + 10 q^{86} + 3 q^{88} - 16 q^{89} - 22 q^{91} + 19 q^{92} + 26 q^{93} - 10 q^{94} + 5 q^{96} + 14 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100})$$ 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 - 5 * q^11 + 7 * q^12 - q^13 - 3 * q^16 + 3 * q^17 + 7 * q^18 + 5 * q^19 + 11 * q^21 - 3 * q^22 + 8 * q^23 + 9 * q^24 - 16 * q^26 + 10 * q^27 + 22 * q^28 + 11 * q^29 - 5 * q^31 + 2 * q^32 - 7 * q^33 + 4 * q^34 - 3 * q^36 + 9 * q^37 + 3 * q^38 - 8 * q^39 + 15 * q^41 - 11 * q^42 + 13 * q^43 - 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^48 + 20 * q^49 + 24 * q^51 - q^52 + 5 * q^53 + 17 * q^54 + 7 * q^57 + 33 * q^58 - 17 * q^59 + 3 * q^61 - 14 * q^62 + 22 * q^63 - 17 * q^64 - 2 * q^66 + 28 * q^67 + 25 * q^68 - 2 * q^69 - 6 * q^71 + 26 * q^72 + 16 * q^73 - 21 * q^74 + 5 * q^76 - 11 * q^77 - 29 * q^78 + 3 * q^79 + q^81 - 2 * q^82 + 33 * q^83 + 33 * q^84 + 10 * q^86 + 3 * q^88 - 16 * q^89 - 22 * q^91 + 19 * q^92 + 26 * q^93 - 10 * q^94 + 5 * q^96 + 14 * q^97 - 10 * q^98 - 8 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{22} + \zeta_{22}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 4\nu^{2} + 2$$ v^4 - 4*v^2 + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 4\beta_{2} + 6$$ b4 + 4*b2 + 6

## 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
 0.284630 −1.68251 −0.830830 1.30972 1.91899
−2.22871 1.47889 2.96714 0 −3.29602 4.42518 −2.15546 −0.812880 0
1.2 −0.0881559 3.20362 −1.99223 0 −0.282418 1.95185 0.351939 7.26315 0
1.3 1.37279 1.23648 −0.115460 0 1.69742 −2.43232 −2.90407 −1.47112 0
1.4 1.54620 −1.51334 0.390736 0 −2.33992 4.16140 −2.48825 −0.709811 0
1.5 2.39788 2.59435 3.74982 0 6.22094 2.89389 4.19584 3.73066 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$11$$ $$+1$$
$$19$$ $$-1$$

## 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 5225.2.a.j 5
5.b even 2 1 1045.2.a.d 5
15.d odd 2 1 9405.2.a.v 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.d 5 5.b even 2 1
5225.2.a.j 5 1.a even 1 1 trivial
9405.2.a.v 5 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{5} - 3T_{2}^{4} - 3T_{2}^{3} + 15T_{2}^{2} - 10T_{2} - 1$$ T2^5 - 3*T2^4 - 3*T2^3 + 15*T2^2 - 10*T2 - 1 $$T_{7}^{5} - 11T_{7}^{4} + 33T_{7}^{3} + 22T_{7}^{2} - 231T_{7} + 253$$ T7^5 - 11*T7^4 + 33*T7^3 + 22*T7^2 - 231*T7 + 253

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 3 T^{4} + \cdots - 1$$
$3$ $$T^{5} - 7 T^{4} + \cdots + 23$$
$5$ $$T^{5}$$
$7$ $$T^{5} - 11 T^{4} + \cdots + 253$$
$11$ $$(T + 1)^{5}$$
$13$ $$T^{5} + T^{4} + \cdots + 529$$
$17$ $$T^{5} - 3 T^{4} + \cdots - 23$$
$19$ $$(T - 1)^{5}$$
$23$ $$T^{5} - 8 T^{4} + \cdots + 1$$
$29$ $$T^{5} - 11 T^{4} + \cdots + 1441$$
$31$ $$T^{5} + 5 T^{4} + \cdots + 10649$$
$37$ $$T^{5} - 9 T^{4} + \cdots - 3917$$
$41$ $$T^{5} - 15 T^{4} + \cdots + 593$$
$43$ $$T^{5} - 13 T^{4} + \cdots - 28753$$
$47$ $$T^{5} - 20 T^{4} + \cdots + 989$$
$53$ $$T^{5} - 5 T^{4} + \cdots - 3917$$
$59$ $$T^{5} + 17 T^{4} + \cdots + 197$$
$61$ $$T^{5} - 3 T^{4} + \cdots + 241$$
$67$ $$T^{5} - 28 T^{4} + \cdots + 11881$$
$71$ $$T^{5} + 6 T^{4} + \cdots - 23$$
$73$ $$T^{5} - 16 T^{4} + \cdots - 23$$
$79$ $$T^{5} - 3 T^{4} + \cdots - 15203$$
$83$ $$T^{5} - 33 T^{4} + \cdots + 737$$
$89$ $$T^{5} + 16 T^{4} + \cdots + 9791$$
$97$ $$T^{5} - 14 T^{4} + \cdots - 3323$$