# Properties

 Label 9025.2.a.t Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $1$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + ( - 2 \beta + 1) q^{3} + 3 \beta q^{4} + ( - 3 \beta - 1) q^{6} + ( - 2 \beta - 1) q^{7} + (4 \beta + 1) q^{8} + 2 q^{9}+O(q^{10})$$ q + (b + 1) * q^2 + (-2*b + 1) * q^3 + 3*b * q^4 + (-3*b - 1) * q^6 + (-2*b - 1) * q^7 + (4*b + 1) * q^8 + 2 * q^9 $$q + (\beta + 1) q^{2} + ( - 2 \beta + 1) q^{3} + 3 \beta q^{4} + ( - 3 \beta - 1) q^{6} + ( - 2 \beta - 1) q^{7} + (4 \beta + 1) q^{8} + 2 q^{9} + (4 \beta - 1) q^{11} + ( - 3 \beta - 6) q^{12} - 2 q^{13} + ( - 5 \beta - 3) q^{14} + (3 \beta + 5) q^{16} - 3 \beta q^{17} + (2 \beta + 2) q^{18} + (4 \beta + 3) q^{21} + (7 \beta + 3) q^{22} + ( - 2 \beta + 2) q^{23} + ( - 6 \beta - 7) q^{24} + ( - 2 \beta - 2) q^{26} + (2 \beta - 1) q^{27} + ( - 9 \beta - 6) q^{28} + 6 q^{29} + ( - 3 \beta + 3) q^{31} + (3 \beta + 6) q^{32} + ( - 2 \beta - 9) q^{33} + ( - 6 \beta - 3) q^{34} + 6 \beta q^{36} + ( - 3 \beta + 9) q^{37} + (4 \beta - 2) q^{39} + (\beta - 8) q^{41} + (11 \beta + 7) q^{42} + (9 \beta - 4) q^{43} + (9 \beta + 12) q^{44} - 2 \beta q^{46} + ( - \beta - 6) q^{47} + ( - 13 \beta - 1) q^{48} + (8 \beta - 2) q^{49} + (3 \beta + 6) q^{51} - 6 \beta q^{52} + (\beta - 8) q^{53} + (3 \beta + 1) q^{54} + ( - 14 \beta - 9) q^{56} + (6 \beta + 6) q^{58} + (2 \beta - 6) q^{59} + ( - 9 \beta + 9) q^{61} - 3 \beta q^{62} + ( - 4 \beta - 2) q^{63} + (6 \beta - 1) q^{64} + ( - 13 \beta - 11) q^{66} + ( - 6 \beta + 1) q^{67} + ( - 9 \beta - 9) q^{68} + ( - 2 \beta + 6) q^{69} + (4 \beta - 11) q^{71} + (8 \beta + 2) q^{72} + ( - 6 \beta - 1) q^{73} + (3 \beta + 6) q^{74} + ( - 10 \beta - 7) q^{77} + (6 \beta + 2) q^{78} + q^{79} - 11 q^{81} + ( - 6 \beta - 7) q^{82} + (5 \beta - 7) q^{83} + (21 \beta + 12) q^{84} + (14 \beta + 5) q^{86} + ( - 12 \beta + 6) q^{87} + (16 \beta + 15) q^{88} + (6 \beta - 6) q^{89} + (4 \beta + 2) q^{91} - 6 q^{92} + ( - 3 \beta + 9) q^{93} + ( - 8 \beta - 7) q^{94} - 15 \beta q^{96} + ( - \beta + 9) q^{97} + (14 \beta + 6) q^{98} + (8 \beta - 2) q^{99} +O(q^{100})$$ q + (b + 1) * q^2 + (-2*b + 1) * q^3 + 3*b * q^4 + (-3*b - 1) * q^6 + (-2*b - 1) * q^7 + (4*b + 1) * q^8 + 2 * q^9 + (4*b - 1) * q^11 + (-3*b - 6) * q^12 - 2 * q^13 + (-5*b - 3) * q^14 + (3*b + 5) * q^16 - 3*b * q^17 + (2*b + 2) * q^18 + (4*b + 3) * q^21 + (7*b + 3) * q^22 + (-2*b + 2) * q^23 + (-6*b - 7) * q^24 + (-2*b - 2) * q^26 + (2*b - 1) * q^27 + (-9*b - 6) * q^28 + 6 * q^29 + (-3*b + 3) * q^31 + (3*b + 6) * q^32 + (-2*b - 9) * q^33 + (-6*b - 3) * q^34 + 6*b * q^36 + (-3*b + 9) * q^37 + (4*b - 2) * q^39 + (b - 8) * q^41 + (11*b + 7) * q^42 + (9*b - 4) * q^43 + (9*b + 12) * q^44 - 2*b * q^46 + (-b - 6) * q^47 + (-13*b - 1) * q^48 + (8*b - 2) * q^49 + (3*b + 6) * q^51 - 6*b * q^52 + (b - 8) * q^53 + (3*b + 1) * q^54 + (-14*b - 9) * q^56 + (6*b + 6) * q^58 + (2*b - 6) * q^59 + (-9*b + 9) * q^61 - 3*b * q^62 + (-4*b - 2) * q^63 + (6*b - 1) * q^64 + (-13*b - 11) * q^66 + (-6*b + 1) * q^67 + (-9*b - 9) * q^68 + (-2*b + 6) * q^69 + (4*b - 11) * q^71 + (8*b + 2) * q^72 + (-6*b - 1) * q^73 + (3*b + 6) * q^74 + (-10*b - 7) * q^77 + (6*b + 2) * q^78 + q^79 - 11 * q^81 + (-6*b - 7) * q^82 + (5*b - 7) * q^83 + (21*b + 12) * q^84 + (14*b + 5) * q^86 + (-12*b + 6) * q^87 + (16*b + 15) * q^88 + (6*b - 6) * q^89 + (4*b + 2) * q^91 - 6 * q^92 + (-3*b + 9) * q^93 + (-8*b - 7) * q^94 - 15*b * q^96 + (-b + 9) * q^97 + (14*b + 6) * q^98 + (8*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} - 4 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 - 4 * q^7 + 6 * q^8 + 4 * q^9 $$2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} - 4 q^{7} + 6 q^{8} + 4 q^{9} + 2 q^{11} - 15 q^{12} - 4 q^{13} - 11 q^{14} + 13 q^{16} - 3 q^{17} + 6 q^{18} + 10 q^{21} + 13 q^{22} + 2 q^{23} - 20 q^{24} - 6 q^{26} - 21 q^{28} + 12 q^{29} + 3 q^{31} + 15 q^{32} - 20 q^{33} - 12 q^{34} + 6 q^{36} + 15 q^{37} - 15 q^{41} + 25 q^{42} + q^{43} + 33 q^{44} - 2 q^{46} - 13 q^{47} - 15 q^{48} + 4 q^{49} + 15 q^{51} - 6 q^{52} - 15 q^{53} + 5 q^{54} - 32 q^{56} + 18 q^{58} - 10 q^{59} + 9 q^{61} - 3 q^{62} - 8 q^{63} + 4 q^{64} - 35 q^{66} - 4 q^{67} - 27 q^{68} + 10 q^{69} - 18 q^{71} + 12 q^{72} - 8 q^{73} + 15 q^{74} - 24 q^{77} + 10 q^{78} + 2 q^{79} - 22 q^{81} - 20 q^{82} - 9 q^{83} + 45 q^{84} + 24 q^{86} + 46 q^{88} - 6 q^{89} + 8 q^{91} - 12 q^{92} + 15 q^{93} - 22 q^{94} - 15 q^{96} + 17 q^{97} + 26 q^{98} + 4 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 - 4 * q^7 + 6 * q^8 + 4 * q^9 + 2 * q^11 - 15 * q^12 - 4 * q^13 - 11 * q^14 + 13 * q^16 - 3 * q^17 + 6 * q^18 + 10 * q^21 + 13 * q^22 + 2 * q^23 - 20 * q^24 - 6 * q^26 - 21 * q^28 + 12 * q^29 + 3 * q^31 + 15 * q^32 - 20 * q^33 - 12 * q^34 + 6 * q^36 + 15 * q^37 - 15 * q^41 + 25 * q^42 + q^43 + 33 * q^44 - 2 * q^46 - 13 * q^47 - 15 * q^48 + 4 * q^49 + 15 * q^51 - 6 * q^52 - 15 * q^53 + 5 * q^54 - 32 * q^56 + 18 * q^58 - 10 * q^59 + 9 * q^61 - 3 * q^62 - 8 * q^63 + 4 * q^64 - 35 * q^66 - 4 * q^67 - 27 * q^68 + 10 * q^69 - 18 * q^71 + 12 * q^72 - 8 * q^73 + 15 * q^74 - 24 * q^77 + 10 * q^78 + 2 * q^79 - 22 * q^81 - 20 * q^82 - 9 * q^83 + 45 * q^84 + 24 * q^86 + 46 * q^88 - 6 * q^89 + 8 * q^91 - 12 * q^92 + 15 * q^93 - 22 * q^94 - 15 * q^96 + 17 * q^97 + 26 * q^98 + 4 * 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
 −0.618034 1.61803
0.381966 2.23607 −1.85410 0 0.854102 0.236068 −1.47214 2.00000 0
1.2 2.61803 −2.23607 4.85410 0 −5.85410 −4.23607 7.47214 2.00000 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

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 9025.2.a.t yes 2
5.b even 2 1 9025.2.a.m yes 2
19.b odd 2 1 9025.2.a.l 2
95.d odd 2 1 9025.2.a.u yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9025.2.a.l 2 19.b odd 2 1
9025.2.a.m yes 2 5.b even 2 1
9025.2.a.t yes 2 1.a even 1 1 trivial
9025.2.a.u yes 2 95.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2}^{2} - 3T_{2} + 1$$ T2^2 - 3*T2 + 1 $$T_{3}^{2} - 5$$ T3^2 - 5 $$T_{7}^{2} + 4T_{7} - 1$$ T7^2 + 4*T7 - 1 $$T_{11}^{2} - 2T_{11} - 19$$ T11^2 - 2*T11 - 19 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 1$$
$3$ $$T^{2} - 5$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4T - 1$$
$11$ $$T^{2} - 2T - 19$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} + 3T - 9$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 2T - 4$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 3T - 9$$
$37$ $$T^{2} - 15T + 45$$
$41$ $$T^{2} + 15T + 55$$
$43$ $$T^{2} - T - 101$$
$47$ $$T^{2} + 13T + 41$$
$53$ $$T^{2} + 15T + 55$$
$59$ $$T^{2} + 10T + 20$$
$61$ $$T^{2} - 9T - 81$$
$67$ $$T^{2} + 4T - 41$$
$71$ $$T^{2} + 18T + 61$$
$73$ $$T^{2} + 8T - 29$$
$79$ $$(T - 1)^{2}$$
$83$ $$T^{2} + 9T - 11$$
$89$ $$T^{2} + 6T - 36$$
$97$ $$T^{2} - 17T + 71$$