# Properties

 Label 5225.2.a.i Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $1$ 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: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.36497.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1$$ x^5 - 2*x^4 - 3*x^3 + 5*x^2 + 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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{4} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{7}+ \cdots + (\beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q - b4 * q^2 + (-b1 + 1) * q^3 + b3 * q^4 + (-b4 - b2 - b1 - 1) * q^6 + (-b4 + b3 - b2 - b1 - 1) * q^7 + (b4 + b2) * q^8 + (b2 - b1) * q^9 $$q - \beta_{4} q^{2} + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{4} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{7}+ \cdots + (\beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q - b4 * q^2 + (-b1 + 1) * q^3 + b3 * q^4 + (-b4 - b2 - b1 - 1) * q^6 + (-b4 + b3 - b2 - b1 - 1) * q^7 + (b4 + b2) * q^8 + (b2 - b1) * q^9 + q^11 + (-b2 - 1) * q^12 + (b4 + 2*b3 + b1) * q^13 + (-b4 - 2*b1 + 1) * q^14 + (b4 - 2*b3 + b1 - 2) * q^16 + (3*b4 - 2*b3 + 3*b1 + 2) * q^17 + (b4 + b3 - b2 - 1) * q^18 + q^19 + (-2*b4 + b3 - 2*b2 - 2) * q^21 - b4 * q^22 + (b4 - 2*b3 + 3*b2 + 2*b1 + 1) * q^23 + (2*b4 - b3 + 2*b2 + b1 + 2) * q^24 + (-2*b4 - b3 + 3*b2 + b1 - 1) * q^26 + (b4 - b3 + 2*b2 + 3*b1) * q^27 + (b4 - b3 + 2) * q^28 + (3*b4 - 4*b3 - 2*b2 + 4*b1 - 4) * q^29 + (b3 + 3*b2 + 3*b1 - 2) * q^31 + (2*b4 - b3 - 3*b2 + b1 - 1) * q^32 + (-b1 + 1) * q^33 + (-3*b3 + b2 + 3*b1 - 3) * q^34 + (-b4 - 2*b3 - b2 + b1 - 2) * q^36 + (-4*b3 + 4*b1 - 3) * q^37 - b4 * q^38 + (b4 - 2*b2 + b1 - 3) * q^39 + (-5*b4 + 3*b3 - 2*b2 - 5*b1 - 5) * q^41 + (-b4 + b2 - 2*b1 + 4) * q^42 + (3*b4 + 2*b3 + 5*b2 + b1) * q^43 + b3 * q^44 + (4*b4 + 2*b3 + 5*b1) * q^46 + (-3*b4 - 3*b3 - 2*b2 - 4*b1 + 2) * q^47 + (b4 + 2*b2 + 3*b1 - 1) * q^48 + (-b3 + b2 - 2*b1 - 1) * q^49 + (3*b4 + 2*b2 + b1 + 1) * q^51 + (3*b4 + b3 + 2*b1 + 5) * q^52 + (b4 - 2*b3 + 3*b1 + 2) * q^53 + (3*b4 + b3 + 2*b2 + 5*b1 + 1) * q^54 + (b4 - b3 - b2 + 4*b1 - 4) * q^56 + (-b1 + 1) * q^57 + (6*b4 - 5*b3 + 2*b1 - 2) * q^58 + (3*b4 - 2*b2 - b1 - 4) * q^59 + (-4*b4 - 2*b3 - 3*b2 - 2*b1 - 7) * q^61 + (4*b4 + 3*b3 + 4*b2 + 6*b1 + 3) * q^62 + (-b4 - b3 - 2*b2 + 3*b1 - 4) * q^63 + (-3*b4 - b3 - 4*b1 + 1) * q^64 + (-b4 - b2 - b1 - 1) * q^66 + (-4*b4 - b3 - 3*b2 - 7*b1 + 1) * q^67 + (b4 + 5*b3 - 2*b1 - 1) * q^68 + (4*b4 - 3*b3 + 4*b2 + 3) * q^69 + (-2*b4 + b3 + 2*b2 - b1 - 5) * q^71 + (b4 - 2*b3 + b2 + 5) * q^72 + (-2*b4 - 2*b3 - 7*b2) * q^73 + (7*b4 + 4*b1 + 4) * q^74 + b3 * q^76 + (-b4 + b3 - b2 - b1 - 1) * q^77 + (b4 - 3*b3 + b2 - b1 - 1) * q^78 + (-2*b4 + 2*b3 + 2*b2 - 6*b1 + 3) * q^79 + (3*b4 - 2*b3 - 2*b2 + 4*b1 - 2) * q^81 + (3*b3 - 2*b2 - 7*b1 + 5) * q^82 + (-b4 + 6*b3 - 7*b1 + 8) * q^83 + (b4 + 2*b2 - b1 + 4) * q^84 + (3*b4 + 2*b3 + 3*b2 + 6*b1 - 5) * q^86 + (b4 + 2*b3 + b2 + 7*b1 - 7) * q^87 + (b4 + b2) * q^88 + (3*b4 + 4*b3 - 3*b1 - 1) * q^89 + (4*b4 - 2*b3 + b2 + b1 + 4) * q^91 + (-4*b4 + b2 + b1 - 5) * q^92 + (3*b4 - 3*b3 - b2 + 2*b1 - 6) * q^93 + (-b4 + b3 - 7*b2 - 6*b1 + 2) * q^94 + (-b4 + 3*b3 - b2 + 3*b1 - 3) * q^96 + (-b4 - 2*b3 + 2*b2 + 3) * q^97 + (3*b4 + b3 - 3*b2 - b1 - 2) * q^98 + (b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + q^{2} + 3 q^{3} + q^{4} - 4 q^{6} - 3 q^{7} - 3 q^{8} - 4 q^{9}+O(q^{10})$$ 5 * q + q^2 + 3 * q^3 + q^4 - 4 * q^6 - 3 * q^7 - 3 * q^8 - 4 * q^9 $$5 q + q^{2} + 3 q^{3} + q^{4} - 4 q^{6} - 3 q^{7} - 3 q^{8} - 4 q^{9} + 5 q^{11} - 3 q^{12} + 3 q^{13} + 2 q^{14} - 11 q^{16} + 11 q^{17} - 3 q^{18} + 5 q^{19} - 3 q^{21} + q^{22} + 5 q^{24} - 8 q^{26} + 8 q^{28} - 15 q^{29} - 9 q^{31} + 3 q^{33} - 14 q^{34} - 7 q^{36} - 11 q^{37} + q^{38} - 10 q^{39} - 23 q^{41} + 15 q^{42} - 9 q^{43} + q^{44} + 8 q^{46} + 6 q^{47} - 4 q^{48} - 12 q^{49} + 27 q^{52} + 13 q^{53} + 9 q^{54} - 12 q^{56} + 3 q^{57} - 17 q^{58} - 21 q^{59} - 31 q^{61} + 18 q^{62} - 10 q^{63} - q^{64} - 4 q^{66} - 5 q^{68} - 28 q^{71} + 20 q^{72} + 14 q^{73} + 21 q^{74} + q^{76} - 3 q^{77} - 13 q^{78} + 3 q^{79} - 3 q^{81} + 18 q^{82} + 33 q^{83} + 13 q^{84} - 20 q^{86} - 22 q^{87} - 3 q^{88} - 10 q^{89} + 14 q^{91} - 21 q^{92} - 30 q^{93} + 14 q^{94} - 3 q^{96} + 10 q^{97} - 8 q^{98} - 4 q^{99}+O(q^{100})$$ 5 * q + q^2 + 3 * q^3 + q^4 - 4 * q^6 - 3 * q^7 - 3 * q^8 - 4 * q^9 + 5 * q^11 - 3 * q^12 + 3 * q^13 + 2 * q^14 - 11 * q^16 + 11 * q^17 - 3 * q^18 + 5 * q^19 - 3 * q^21 + q^22 + 5 * q^24 - 8 * q^26 + 8 * q^28 - 15 * q^29 - 9 * q^31 + 3 * q^33 - 14 * q^34 - 7 * q^36 - 11 * q^37 + q^38 - 10 * q^39 - 23 * q^41 + 15 * q^42 - 9 * q^43 + q^44 + 8 * q^46 + 6 * q^47 - 4 * q^48 - 12 * q^49 + 27 * q^52 + 13 * q^53 + 9 * q^54 - 12 * q^56 + 3 * q^57 - 17 * q^58 - 21 * q^59 - 31 * q^61 + 18 * q^62 - 10 * q^63 - q^64 - 4 * q^66 - 5 * q^68 - 28 * q^71 + 20 * q^72 + 14 * q^73 + 21 * q^74 + q^76 - 3 * q^77 - 13 * q^78 + 3 * q^79 - 3 * q^81 + 18 * q^82 + 33 * q^83 + 13 * q^84 - 20 * q^86 - 22 * q^87 - 3 * q^88 - 10 * q^89 + 14 * q^91 - 21 * q^92 - 30 * q^93 + 14 * q^94 - 3 * q^96 + 10 * q^97 - 8 * q^98 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2$$ v^4 - v^3 - 4*v^2 + 2*v + 2 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1$$ v^4 - 2*v^3 - 3*v^2 + 4*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 1$$ -b4 + b3 + b2 + 3*b1 + 1 $$\nu^{4}$$ $$=$$ $$-\beta_{4} + 2\beta_{3} + 5\beta_{2} + 5\beta _1 + 7$$ -b4 + 2*b3 + 5*b2 + 5*b1 + 7

## 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
 0.410375 −1.55629 1.33419 −0.506287 2.31801
−2.02642 0.589625 2.10637 0 −1.19483 0.911544 −0.215549 −2.65234 0
1.2 −0.913732 2.55629 −1.16509 0 −2.33576 −3.50085 2.89205 3.53460 0
1.3 0.584664 −0.334185 −1.65817 0 −0.195386 −1.85355 −2.13880 −2.88832 0
1.4 1.46888 1.50629 0.157597 0 2.21255 2.37015 −2.70626 −0.731099 0
1.5 1.88661 −1.31801 1.55930 0 −2.48658 −0.927281 −0.831437 −1.26284 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.i 5
5.b even 2 1 1045.2.a.e 5
15.d odd 2 1 9405.2.a.u 5

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

 $$T_{2}^{5} - T_{2}^{4} - 5T_{2}^{3} + 5T_{2}^{2} + 4T_{2} - 3$$ T2^5 - T2^4 - 5*T2^3 + 5*T2^2 + 4*T2 - 3 $$T_{7}^{5} + 3T_{7}^{4} - 7T_{7}^{3} - 18T_{7}^{2} + 5T_{7} + 13$$ T7^5 + 3*T7^4 - 7*T7^3 - 18*T7^2 + 5*T7 + 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - T^{4} - 5 T^{3} + \cdots - 3$$
$3$ $$T^{5} - 3 T^{4} + \cdots - 1$$
$5$ $$T^{5}$$
$7$ $$T^{5} + 3 T^{4} + \cdots + 13$$
$11$ $$(T - 1)^{5}$$
$13$ $$T^{5} - 3 T^{4} + \cdots + 299$$
$17$ $$T^{5} - 11 T^{4} + \cdots + 69$$
$19$ $$(T - 1)^{5}$$
$23$ $$T^{5} - 74 T^{3} + \cdots - 1821$$
$29$ $$T^{5} + 15 T^{4} + \cdots - 1047$$
$31$ $$T^{5} + 9 T^{4} + \cdots - 3025$$
$37$ $$T^{5} + 11 T^{4} + \cdots + 79$$
$41$ $$T^{5} + 23 T^{4} + \cdots - 22467$$
$43$ $$T^{5} + 9 T^{4} + \cdots + 2783$$
$47$ $$T^{5} - 6 T^{4} + \cdots - 21621$$
$53$ $$T^{5} - 13 T^{4} + \cdots + 27$$
$59$ $$T^{5} + 21 T^{4} + \cdots - 6849$$
$61$ $$T^{5} + 31 T^{4} + \cdots + 3307$$
$67$ $$T^{5} - 195 T^{3} + \cdots - 1031$$
$71$ $$T^{5} + 28 T^{4} + \cdots - 141$$
$73$ $$T^{5} - 14 T^{4} + \cdots - 115699$$
$79$ $$T^{5} - 3 T^{4} + \cdots + 6125$$
$83$ $$T^{5} - 33 T^{4} + \cdots + 77949$$
$89$ $$T^{5} + 10 T^{4} + \cdots + 6909$$
$97$ $$T^{5} - 10 T^{4} + \cdots + 1597$$