# Properties

 Label 5225.2.a.k Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.131947641.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 10x^{4} + 25x^{2} - 3x - 9$$ x^6 - 10*x^4 + 25*x^2 - 3*x - 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{3} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2) q^{4} + ( - \beta_{5} + \beta_1) q^{6} + (\beta_{4} - \beta_1 - 1) q^{7} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b3 - 1) * q^3 + (-b3 + b2 + 2) * q^4 + (-b5 + b1) * q^6 + (b4 - b1 - 1) * q^7 + (-b4 - b2 - b1) * q^8 + (-b4 - b3 - b2 + 2) * q^9 $$q - \beta_1 q^{2} + (\beta_{3} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2) q^{4} + ( - \beta_{5} + \beta_1) q^{6} + (\beta_{4} - \beta_1 - 1) q^{7} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{9} - q^{11} + ( - \beta_{5} + \beta_{3} - 4) q^{12} + ( - \beta_{3} + \beta_1 + 2) q^{13} + (\beta_1 + 3) q^{14} + (\beta_{5} + \beta_{4} + \beta_{2} + \cdots + 1) q^{16}+ \cdots + (\beta_{4} + \beta_{3} + \beta_{2} - 2) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b3 - 1) * q^3 + (-b3 + b2 + 2) * q^4 + (-b5 + b1) * q^6 + (b4 - b1 - 1) * q^7 + (-b4 - b2 - b1) * q^8 + (-b4 - b3 - b2 + 2) * q^9 - q^11 + (-b5 + b3 - 4) * q^12 + (-b3 + b1 + 2) * q^13 + (b1 + 3) * q^14 + (b5 + b4 + b2 + b1 + 1) * q^16 + (b3 - 2*b2 + b1) * q^17 + (2*b5 + b4 - b3 + 2*b2 - b1 + 1) * q^18 - q^19 + (-b5 + b4 - 2*b3 + 1) * q^21 + b1 * q^22 + (-b4 + b2 + b1 - 1) * q^23 + (2*b3 + b2 + 2*b1 - 2) * q^24 + (b5 + b3 - b2 - 2*b1 - 4) * q^26 + (b5 + b4 + b3 + 2*b2 + b1 - 5) * q^27 + (-2*b4 + b3 - b2 - b1 - 2) * q^28 + (b5 - b4 + b3 - b2 + b1 - 1) * q^29 + (-b4 - 2*b1) * q^31 + (b4 - 2*b2 - 3) * q^32 + (-b3 + 1) * q^33 + (b5 + 2*b4 + b3 + b2 + 2*b1 - 4) * q^34 + (b5 - 2*b3 - 2*b2 - 3*b1 + 3) * q^36 + (-b5 - b4 + b3 + b2 - 2) * q^37 + b1 * q^38 + (b5 + b4 + 2*b3 + b2 - b1 - 6) * q^39 + (2*b5 + b2 + b1 + 5) * q^41 + (b5 + 3*b3 - b1 - 3) * q^42 + (-b4 - b3 - b2 - b1 - 2) * q^43 + (b3 - b2 - 2) * q^44 + (-b5 - b4 - b2 - 3) * q^46 + (-2*b5 - b3 + b2 + b1 - 4) * q^47 + (-b5 - b4 - 3*b2 + b1) * q^48 + (b5 + 4*b3 - 3*b2 + 2*b1) * q^49 + (3*b5 + b4 + 2*b3 + b2 - b1) * q^51 + (b5 + b4 - 2*b3 + 2*b2 + 3*b1 + 6) * q^52 + (2*b5 + 4*b4 - b3 + 2*b2 + b1) * q^53 + (-2*b5 - 2*b4 - 5*b2 + 3*b1 - 3) * q^54 + (b4 - 3*b3 + 4*b2 + b1) * q^56 + (-b3 + 1) * q^57 + (b5 + b4 - 2*b3 + 2*b1 - 1) * q^58 + (b3 + b1 + 6) * q^59 + (-b5 + 2*b4 - b3 + 2*b1 - 2) * q^61 + (-3*b3 + 3*b2 + 9) * q^62 + (b5 + b4 - b3 + 4*b2 - b1 - 5) * q^63 + (b3 - b2 + 3*b1 - 3) * q^64 + (b5 - b1) * q^66 + (-b5 - 2*b4 - b3 - 3*b2 - 4) * q^67 + (-b5 - b4 - 2*b2 + b1 - 8) * q^68 + (-2*b4 - b3 - b2 + 3) * q^69 + (-3*b5 - b4 - b3 + 2*b2 - 2*b1 - 4) * q^71 + (b5 - 3*b3 + b1 + 12) * q^72 + (-b4 + 2*b3 + 3*b2 + 4) * q^73 + (-3*b5 - b4 + b3 + b2 + b1 - 1) * q^74 + (b3 - b2 - 2) * q^76 + (-b4 + b1 + 1) * q^77 + (-2*b5 - b4 - 2*b3 - 2*b2 + 5*b1 + 5) * q^78 + (-b5 + b4 - b3 + b2 + 2) * q^79 + (-2*b5 - 4*b3 - 2*b2 + b1 + 6) * q^81 + (b5 - b4 - 3*b3 - 4*b2 - 6*b1) * q^82 + (2*b5 + 2*b4 - 3*b3 + 4*b2 + b1 - 4) * q^83 + (-2*b4 + b3 + 3*b1 + 4) * q^84 + (2*b5 + b4 - 2*b3 + 3*b2 + 3*b1 + 5) * q^86 + (b5 - 2*b4 + 2*b3 - 2*b2 + 3*b1 + 2) * q^87 + (b4 + b2 + b1) * q^88 + (-b5 + b4 - 2*b3 - b2 + b1 + 4) * q^89 + (b5 + 2*b3 - 2*b1 - 5) * q^91 + (3*b4 + b3 + b2 + 2*b1 + 1) * q^92 + (-2*b5 - b4 + b3 + 3*b1) * q^93 + (-2*b5 - b4 + 5*b3 + 3*b1 - 8) * q^94 + (2*b5 + 3*b4 - 2*b3 + 2*b2 - b1 - 1) * q^96 + (2*b5 + 2*b4 + 2*b3 - 4*b2 - b1 - 5) * q^97 + (3*b4 + 3*b1 - 6) * q^98 + (b4 + b3 + b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9}+O(q^{10})$$ 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 $$6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28} - 5 q^{29} - q^{31} - 15 q^{32} + 3 q^{33} - 22 q^{34} + 12 q^{36} - 9 q^{37} - 32 q^{39} + 25 q^{41} - 11 q^{42} - 15 q^{43} - 8 q^{44} - 16 q^{46} - 24 q^{47} + 4 q^{48} + 13 q^{49} + 27 q^{52} - 5 q^{53} - 11 q^{54} - 12 q^{56} + 3 q^{57} - 13 q^{58} + 39 q^{59} - 11 q^{61} + 42 q^{62} - 38 q^{63} - 14 q^{64} - 2 q^{66} - 24 q^{67} - 45 q^{68} + 14 q^{69} - 24 q^{71} + 61 q^{72} + 26 q^{73} + q^{74} - 8 q^{76} + 5 q^{77} + 29 q^{78} + 11 q^{79} + 30 q^{81} - 8 q^{82} - 39 q^{83} + 25 q^{84} + 18 q^{86} + 16 q^{87} + 22 q^{89} - 26 q^{91} + 11 q^{92} + 6 q^{93} - 30 q^{94} - 15 q^{96} - 22 q^{97} - 33 q^{98} - 9 q^{99}+O(q^{100})$$ 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 10x^{4} + 25x^{2} - 3x - 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 7\nu^{3} + 7\nu - 3 ) / 3$$ (v^5 - 7*v^3 + 7*v - 3) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 7\nu^{3} - 3\nu^{2} + 7\nu + 9 ) / 3$$ (v^5 - 7*v^3 - 3*v^2 + 7*v + 9) / 3 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 10\nu^{3} - 22\nu + 3 ) / 3$$ (-v^5 + 10*v^3 - 22*v + 3) / 3 $$\beta_{5}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 3$$ v^4 - v^3 - 6*v^2 + 4*v + 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 4$$ -b3 + b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{2} + 5\beta_1$$ b4 + b2 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{4} - 6\beta_{3} + 7\beta_{2} + \beta _1 + 21$$ b5 + b4 - 6*b3 + 7*b2 + b1 + 21 $$\nu^{5}$$ $$=$$ $$7\beta_{4} + 10\beta_{2} + 28\beta _1 + 3$$ 7*b4 + 10*b2 + 28*b1 + 3

## 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
 2.56745 1.65636 0.759131 −0.577704 −2.04201 −2.36323
−2.56745 −0.903918 4.59179 0 2.32076 −2.16848 −6.65430 −2.18293 0
1.2 −1.65636 −3.32622 0.743534 0 5.50942 −2.81120 2.08116 8.06374 0
1.3 −0.759131 2.25829 −1.42372 0 −1.71434 −4.95189 2.59905 2.09989 0
1.4 0.577704 0.746709 −1.66626 0 0.431377 4.19297 −2.11801 −2.44243 0
1.5 2.04201 1.09835 2.16980 0 2.24284 0.469142 0.346728 −1.79363 0
1.6 2.36323 −2.87321 3.58485 0 −6.79006 0.269449 3.74537 5.25535 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.k 6
5.b even 2 1 1045.2.a.g 6
15.d odd 2 1 9405.2.a.w 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.g 6 5.b even 2 1
5225.2.a.k 6 1.a even 1 1 trivial
9405.2.a.w 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{6} - 10T_{2}^{4} + 25T_{2}^{2} + 3T_{2} - 9$$ T2^6 - 10*T2^4 + 25*T2^2 + 3*T2 - 9 $$T_{7}^{6} + 5T_{7}^{5} - 15T_{7}^{4} - 90T_{7}^{3} - 55T_{7}^{2} + 81T_{7} - 16$$ T7^6 + 5*T7^5 - 15*T7^4 - 90*T7^3 - 55*T7^2 + 81*T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 10 T^{4} + \cdots - 9$$
$3$ $$T^{6} + 3 T^{5} + \cdots - 16$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 5 T^{5} + \cdots - 16$$
$11$ $$(T + 1)^{6}$$
$13$ $$T^{6} - 9 T^{5} + \cdots - 14$$
$17$ $$T^{6} - 5 T^{5} + \cdots - 6290$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} + 8 T^{5} + \cdots + 92$$
$29$ $$T^{6} + 5 T^{5} + \cdots + 482$$
$31$ $$T^{6} + T^{5} + \cdots - 3240$$
$37$ $$T^{6} + 9 T^{5} + \cdots + 1906$$
$41$ $$T^{6} - 25 T^{5} + \cdots + 9526$$
$43$ $$T^{6} + 15 T^{5} + \cdots + 4460$$
$47$ $$T^{6} + 24 T^{5} + \cdots - 7268$$
$53$ $$T^{6} + 5 T^{5} + \cdots - 627158$$
$59$ $$T^{6} - 39 T^{5} + \cdots + 35420$$
$61$ $$T^{6} + 11 T^{5} + \cdots - 156150$$
$67$ $$T^{6} + 24 T^{5} + \cdots - 123184$$
$71$ $$T^{6} + 24 T^{5} + \cdots - 56840$$
$73$ $$T^{6} - 26 T^{5} + \cdots + 34902$$
$79$ $$T^{6} - 11 T^{5} + \cdots + 8152$$
$83$ $$T^{6} + 39 T^{5} + \cdots + 2100044$$
$89$ $$T^{6} - 22 T^{5} + \cdots + 554$$
$97$ $$T^{6} + 22 T^{5} + \cdots + 506$$