# Properties

 Label 9025.2.a.bl Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 5$$ x^4 - 5*x^2 + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1805) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} + 3) q^{6} + (2 \beta_{2} + 3) q^{7} + (\beta_{3} - \beta_1) q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^2 + b1 * q^3 + (b2 + 1) * q^4 + (b2 + 3) * q^6 + (2*b2 + 3) * q^7 + (b3 - b1) * q^8 + b2 * q^9 $$q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} + 3) q^{6} + (2 \beta_{2} + 3) q^{7} + (\beta_{3} - \beta_1) q^{8} + \beta_{2} q^{9} + ( - 3 \beta_{2} - 4) q^{11} + (\beta_{3} + \beta_1) q^{12} + ( - \beta_{3} - \beta_1) q^{13} + (2 \beta_{3} + 3 \beta_1) q^{14} + ( - \beta_{2} - 4) q^{16} + ( - 2 \beta_{2} - 4) q^{17} + \beta_{3} q^{18} + (2 \beta_{3} + 3 \beta_1) q^{21} + ( - 3 \beta_{3} - 4 \beta_1) q^{22} + ( - 5 \beta_{2} - 1) q^{23} + (\beta_{2} - 2) q^{24} + ( - 3 \beta_{2} - 4) q^{26} + (\beta_{3} - 3 \beta_1) q^{27} + (3 \beta_{2} + 5) q^{28} + ( - 4 \beta_{3} + \beta_1) q^{29} + \beta_1 q^{31} + ( - 3 \beta_{3} - 2 \beta_1) q^{32} + ( - 3 \beta_{3} - 4 \beta_1) q^{33} + ( - 2 \beta_{3} - 4 \beta_1) q^{34} + q^{36} + (4 \beta_{3} - \beta_1) q^{37} + ( - 3 \beta_{2} - 4) q^{39} + ( - \beta_{3} - 3 \beta_1) q^{41} + (7 \beta_{2} + 11) q^{42} + (\beta_{2} - 1) q^{43} + ( - 4 \beta_{2} - 7) q^{44} + ( - 5 \beta_{3} - \beta_1) q^{46} + (4 \beta_{2} - 1) q^{47} + ( - \beta_{3} - 4 \beta_1) q^{48} + (8 \beta_{2} + 6) q^{49} + ( - 2 \beta_{3} - 4 \beta_1) q^{51} + ( - \beta_{3} - 2 \beta_1) q^{52} + ( - 3 \beta_{3} - 4 \beta_1) q^{53} + ( - \beta_{2} - 8) q^{54} + ( - \beta_{3} - \beta_1) q^{56} + ( - 7 \beta_{2} - 1) q^{58} + ( - \beta_{3} + 8 \beta_1) q^{59} + (8 \beta_{2} - 1) q^{61} + (\beta_{2} + 3) q^{62} + (\beta_{2} + 2) q^{63} + ( - 6 \beta_{2} - 1) q^{64} + ( - 10 \beta_{2} - 15) q^{66} + (3 \beta_{3} - 5 \beta_1) q^{67} + ( - 4 \beta_{2} - 6) q^{68} + ( - 5 \beta_{3} - \beta_1) q^{69} + (5 \beta_{3} - 3 \beta_1) q^{71} + ( - 2 \beta_{3} + \beta_1) q^{72} + q^{73} + (7 \beta_{2} + 1) q^{74} + ( - 11 \beta_{2} - 18) q^{77} + ( - 3 \beta_{3} - 4 \beta_1) q^{78} + (2 \beta_{3} - 4 \beta_1) q^{79} + ( - 4 \beta_{2} - 8) q^{81} + ( - 5 \beta_{2} - 10) q^{82} + ( - 2 \beta_{2} + 10) q^{83} + (3 \beta_{3} + 5 \beta_1) q^{84} + (\beta_{3} - \beta_1) q^{86} + ( - 7 \beta_{2} - 1) q^{87} + (2 \beta_{3} + \beta_1) q^{88} + (5 \beta_{3} + \beta_1) q^{89} + ( - 3 \beta_{3} - 5 \beta_1) q^{91} + ( - \beta_{2} - 6) q^{92} + (\beta_{2} + 3) q^{93} + (4 \beta_{3} - \beta_1) q^{94} + ( - 8 \beta_{2} - 9) q^{96} + ( - \beta_{3} + 2 \beta_1) q^{97} + (8 \beta_{3} + 6 \beta_1) q^{98} + ( - \beta_{2} - 3) q^{99}+O(q^{100})$$ q + b1 * q^2 + b1 * q^3 + (b2 + 1) * q^4 + (b2 + 3) * q^6 + (2*b2 + 3) * q^7 + (b3 - b1) * q^8 + b2 * q^9 + (-3*b2 - 4) * q^11 + (b3 + b1) * q^12 + (-b3 - b1) * q^13 + (2*b3 + 3*b1) * q^14 + (-b2 - 4) * q^16 + (-2*b2 - 4) * q^17 + b3 * q^18 + (2*b3 + 3*b1) * q^21 + (-3*b3 - 4*b1) * q^22 + (-5*b2 - 1) * q^23 + (b2 - 2) * q^24 + (-3*b2 - 4) * q^26 + (b3 - 3*b1) * q^27 + (3*b2 + 5) * q^28 + (-4*b3 + b1) * q^29 + b1 * q^31 + (-3*b3 - 2*b1) * q^32 + (-3*b3 - 4*b1) * q^33 + (-2*b3 - 4*b1) * q^34 + q^36 + (4*b3 - b1) * q^37 + (-3*b2 - 4) * q^39 + (-b3 - 3*b1) * q^41 + (7*b2 + 11) * q^42 + (b2 - 1) * q^43 + (-4*b2 - 7) * q^44 + (-5*b3 - b1) * q^46 + (4*b2 - 1) * q^47 + (-b3 - 4*b1) * q^48 + (8*b2 + 6) * q^49 + (-2*b3 - 4*b1) * q^51 + (-b3 - 2*b1) * q^52 + (-3*b3 - 4*b1) * q^53 + (-b2 - 8) * q^54 + (-b3 - b1) * q^56 + (-7*b2 - 1) * q^58 + (-b3 + 8*b1) * q^59 + (8*b2 - 1) * q^61 + (b2 + 3) * q^62 + (b2 + 2) * q^63 + (-6*b2 - 1) * q^64 + (-10*b2 - 15) * q^66 + (3*b3 - 5*b1) * q^67 + (-4*b2 - 6) * q^68 + (-5*b3 - b1) * q^69 + (5*b3 - 3*b1) * q^71 + (-2*b3 + b1) * q^72 + q^73 + (7*b2 + 1) * q^74 + (-11*b2 - 18) * q^77 + (-3*b3 - 4*b1) * q^78 + (2*b3 - 4*b1) * q^79 + (-4*b2 - 8) * q^81 + (-5*b2 - 10) * q^82 + (-2*b2 + 10) * q^83 + (3*b3 + 5*b1) * q^84 + (b3 - b1) * q^86 + (-7*b2 - 1) * q^87 + (2*b3 + b1) * q^88 + (5*b3 + b1) * q^89 + (-3*b3 - 5*b1) * q^91 + (-b2 - 6) * q^92 + (b2 + 3) * q^93 + (4*b3 - b1) * q^94 + (-8*b2 - 9) * q^96 + (-b3 + 2*b1) * q^97 + (8*b3 + 6*b1) * q^98 + (-b2 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 $$4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49} - 30 q^{54} + 10 q^{58} - 20 q^{61} + 10 q^{62} + 6 q^{63} + 8 q^{64} - 40 q^{66} - 16 q^{68} + 4 q^{73} - 10 q^{74} - 50 q^{77} - 24 q^{81} - 30 q^{82} + 44 q^{83} + 10 q^{87} - 22 q^{92} + 10 q^{93} - 20 q^{96} - 10 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 - 12 * q^17 + 6 * q^23 - 10 * q^24 - 10 * q^26 + 14 * q^28 + 4 * q^36 - 10 * q^39 + 30 * q^42 - 6 * q^43 - 20 * q^44 - 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 6 * q^63 + 8 * q^64 - 40 * q^66 - 16 * q^68 + 4 * q^73 - 10 * q^74 - 50 * q^77 - 24 * q^81 - 30 * q^82 + 44 * q^83 + 10 * q^87 - 22 * q^92 + 10 * q^93 - 20 * q^96 - 10 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{20} + \zeta_{20}^{-1}$$:

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

## 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
 −1.90211 −1.17557 1.17557 1.90211
−1.90211 −1.90211 1.61803 0 3.61803 4.23607 0.726543 0.618034 0
1.2 −1.17557 −1.17557 −0.618034 0 1.38197 −0.236068 3.07768 −1.61803 0
1.3 1.17557 1.17557 −0.618034 0 1.38197 −0.236068 −3.07768 −1.61803 0
1.4 1.90211 1.90211 1.61803 0 3.61803 4.23607 −0.726543 0.618034 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
$$5$$ $$1$$
$$19$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9025.2.a.bl 4
5.b even 2 1 1805.2.a.l 4
19.b odd 2 1 inner 9025.2.a.bl 4
95.d odd 2 1 1805.2.a.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.l 4 5.b even 2 1
1805.2.a.l 4 95.d odd 2 1
9025.2.a.bl 4 1.a even 1 1 trivial
9025.2.a.bl 4 19.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2}^{4} - 5T_{2}^{2} + 5$$ T2^4 - 5*T2^2 + 5 $$T_{3}^{4} - 5T_{3}^{2} + 5$$ T3^4 - 5*T3^2 + 5 $$T_{7}^{2} - 4T_{7} - 1$$ T7^2 - 4*T7 - 1 $$T_{11}^{2} + 5T_{11} - 5$$ T11^2 + 5*T11 - 5 $$T_{29}^{4} - 85T_{29}^{2} + 605$$ T29^4 - 85*T29^2 + 605

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 5T^{2} + 5$$
$3$ $$T^{4} - 5T^{2} + 5$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 4 T - 1)^{2}$$
$11$ $$(T^{2} + 5 T - 5)^{2}$$
$13$ $$T^{4} - 10T^{2} + 5$$
$17$ $$(T^{2} + 6 T + 4)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} - 3 T - 29)^{2}$$
$29$ $$T^{4} - 85T^{2} + 605$$
$31$ $$T^{4} - 5T^{2} + 5$$
$37$ $$T^{4} - 85T^{2} + 605$$
$41$ $$T^{4} - 50T^{2} + 125$$
$43$ $$(T^{2} + 3 T + 1)^{2}$$
$47$ $$(T^{2} + 6 T - 11)^{2}$$
$53$ $$T^{4} - 125T^{2} + 125$$
$59$ $$T^{4} - 325 T^{2} + 25205$$
$61$ $$(T^{2} + 10 T - 55)^{2}$$
$67$ $$T^{4} - 170T^{2} + 4805$$
$71$ $$T^{4} - 170T^{2} + 5$$
$73$ $$(T - 1)^{4}$$
$79$ $$T^{4} - 100T^{2} + 2000$$
$83$ $$(T^{2} - 22 T + 116)^{2}$$
$89$ $$T^{4} - 130T^{2} + 4205$$
$97$ $$T^{4} - 25T^{2} + 125$$