# Properties

 Label 9025.2.a.o Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $0$ 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: $$0$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 361) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta + 1) q^{2} - 2 q^{3} + 3 q^{4} + (4 \beta - 2) q^{6} + 2 \beta q^{7} + ( - 2 \beta + 1) q^{8} + q^{9}+O(q^{10})$$ q + (-2*b + 1) * q^2 - 2 * q^3 + 3 * q^4 + (4*b - 2) * q^6 + 2*b * q^7 + (-2*b + 1) * q^8 + q^9 $$q + ( - 2 \beta + 1) q^{2} - 2 q^{3} + 3 q^{4} + (4 \beta - 2) q^{6} + 2 \beta q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} + ( - 2 \beta + 4) q^{11} - 6 q^{12} - 3 \beta q^{13} + ( - 2 \beta - 4) q^{14} - q^{16} + \beta q^{17} + ( - 2 \beta + 1) q^{18} - 4 \beta q^{21} + ( - 6 \beta + 8) q^{22} + ( - 4 \beta + 2) q^{23} + (4 \beta - 2) q^{24} + (3 \beta + 6) q^{26} + 4 q^{27} + 6 \beta q^{28} + ( - 5 \beta + 4) q^{29} - 6 q^{31} + (6 \beta - 3) q^{32} + (4 \beta - 8) q^{33} + ( - \beta - 2) q^{34} + 3 q^{36} + (3 \beta - 7) q^{37} + 6 \beta q^{39} + ( - 5 \beta + 1) q^{41} + (4 \beta + 8) q^{42} + ( - 4 \beta - 2) q^{43} + ( - 6 \beta + 12) q^{44} + 10 q^{46} + ( - 2 \beta - 6) q^{47} + 2 q^{48} + (4 \beta - 3) q^{49} - 2 \beta q^{51} - 9 \beta q^{52} + ( - 5 \beta - 3) q^{53} + ( - 8 \beta + 4) q^{54} + ( - 2 \beta - 4) q^{56} + ( - 3 \beta + 14) q^{58} + ( - 2 \beta + 2) q^{59} + ( - 3 \beta + 8) q^{61} + (12 \beta - 6) q^{62} + 2 \beta q^{63} - 13 q^{64} + (12 \beta - 16) q^{66} + ( - 4 \beta - 2) q^{67} + 3 \beta q^{68} + (8 \beta - 4) q^{69} + (2 \beta - 2) q^{71} + ( - 2 \beta + 1) q^{72} + ( - 9 \beta + 9) q^{73} + (11 \beta - 13) q^{74} + (4 \beta - 4) q^{77} + ( - 6 \beta - 12) q^{78} + 2 q^{79} - 11 q^{81} + (3 \beta + 11) q^{82} + ( - 6 \beta + 6) q^{83} - 12 \beta q^{84} + (8 \beta + 6) q^{86} + (10 \beta - 8) q^{87} + ( - 6 \beta + 8) q^{88} + (7 \beta - 3) q^{89} + ( - 6 \beta - 6) q^{91} + ( - 12 \beta + 6) q^{92} + 12 q^{93} + (14 \beta - 2) q^{94} + ( - 12 \beta + 6) q^{96} + ( - 5 \beta + 8) q^{97} + (2 \beta - 11) q^{98} + ( - 2 \beta + 4) q^{99} +O(q^{100})$$ q + (-2*b + 1) * q^2 - 2 * q^3 + 3 * q^4 + (4*b - 2) * q^6 + 2*b * q^7 + (-2*b + 1) * q^8 + q^9 + (-2*b + 4) * q^11 - 6 * q^12 - 3*b * q^13 + (-2*b - 4) * q^14 - q^16 + b * q^17 + (-2*b + 1) * q^18 - 4*b * q^21 + (-6*b + 8) * q^22 + (-4*b + 2) * q^23 + (4*b - 2) * q^24 + (3*b + 6) * q^26 + 4 * q^27 + 6*b * q^28 + (-5*b + 4) * q^29 - 6 * q^31 + (6*b - 3) * q^32 + (4*b - 8) * q^33 + (-b - 2) * q^34 + 3 * q^36 + (3*b - 7) * q^37 + 6*b * q^39 + (-5*b + 1) * q^41 + (4*b + 8) * q^42 + (-4*b - 2) * q^43 + (-6*b + 12) * q^44 + 10 * q^46 + (-2*b - 6) * q^47 + 2 * q^48 + (4*b - 3) * q^49 - 2*b * q^51 - 9*b * q^52 + (-5*b - 3) * q^53 + (-8*b + 4) * q^54 + (-2*b - 4) * q^56 + (-3*b + 14) * q^58 + (-2*b + 2) * q^59 + (-3*b + 8) * q^61 + (12*b - 6) * q^62 + 2*b * q^63 - 13 * q^64 + (12*b - 16) * q^66 + (-4*b - 2) * q^67 + 3*b * q^68 + (8*b - 4) * q^69 + (2*b - 2) * q^71 + (-2*b + 1) * q^72 + (-9*b + 9) * q^73 + (11*b - 13) * q^74 + (4*b - 4) * q^77 + (-6*b - 12) * q^78 + 2 * q^79 - 11 * q^81 + (3*b + 11) * q^82 + (-6*b + 6) * q^83 - 12*b * q^84 + (8*b + 6) * q^86 + (10*b - 8) * q^87 + (-6*b + 8) * q^88 + (7*b - 3) * q^89 + (-6*b - 6) * q^91 + (-12*b + 6) * q^92 + 12 * q^93 + (14*b - 2) * q^94 + (-12*b + 6) * q^96 + (-5*b + 8) * q^97 + (2*b - 11) * q^98 + (-2*b + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 6 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 6 * q^4 + 2 * q^7 + 2 * q^9 $$2 q - 4 q^{3} + 6 q^{4} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 12 q^{12} - 3 q^{13} - 10 q^{14} - 2 q^{16} + q^{17} - 4 q^{21} + 10 q^{22} + 15 q^{26} + 8 q^{27} + 6 q^{28} + 3 q^{29} - 12 q^{31} - 12 q^{33} - 5 q^{34} + 6 q^{36} - 11 q^{37} + 6 q^{39} - 3 q^{41} + 20 q^{42} - 8 q^{43} + 18 q^{44} + 20 q^{46} - 14 q^{47} + 4 q^{48} - 2 q^{49} - 2 q^{51} - 9 q^{52} - 11 q^{53} - 10 q^{56} + 25 q^{58} + 2 q^{59} + 13 q^{61} + 2 q^{63} - 26 q^{64} - 20 q^{66} - 8 q^{67} + 3 q^{68} - 2 q^{71} + 9 q^{73} - 15 q^{74} - 4 q^{77} - 30 q^{78} + 4 q^{79} - 22 q^{81} + 25 q^{82} + 6 q^{83} - 12 q^{84} + 20 q^{86} - 6 q^{87} + 10 q^{88} + q^{89} - 18 q^{91} + 24 q^{93} + 10 q^{94} + 11 q^{97} - 20 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 + 6 * q^4 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 12 * q^12 - 3 * q^13 - 10 * q^14 - 2 * q^16 + q^17 - 4 * q^21 + 10 * q^22 + 15 * q^26 + 8 * q^27 + 6 * q^28 + 3 * q^29 - 12 * q^31 - 12 * q^33 - 5 * q^34 + 6 * q^36 - 11 * q^37 + 6 * q^39 - 3 * q^41 + 20 * q^42 - 8 * q^43 + 18 * q^44 + 20 * q^46 - 14 * q^47 + 4 * q^48 - 2 * q^49 - 2 * q^51 - 9 * q^52 - 11 * q^53 - 10 * q^56 + 25 * q^58 + 2 * q^59 + 13 * q^61 + 2 * q^63 - 26 * q^64 - 20 * q^66 - 8 * q^67 + 3 * q^68 - 2 * q^71 + 9 * q^73 - 15 * q^74 - 4 * q^77 - 30 * q^78 + 4 * q^79 - 22 * q^81 + 25 * q^82 + 6 * q^83 - 12 * q^84 + 20 * q^86 - 6 * q^87 + 10 * q^88 + q^89 - 18 * q^91 + 24 * q^93 + 10 * q^94 + 11 * q^97 - 20 * q^98 + 6 * 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
 1.61803 −0.618034
−2.23607 −2.00000 3.00000 0 4.47214 3.23607 −2.23607 1.00000 0
1.2 2.23607 −2.00000 3.00000 0 −4.47214 −1.23607 2.23607 1.00000 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.o 2
5.b even 2 1 361.2.a.e yes 2
15.d odd 2 1 3249.2.a.m 2
19.b odd 2 1 9025.2.a.r 2
20.d odd 2 1 5776.2.a.r 2
95.d odd 2 1 361.2.a.d 2
95.h odd 6 2 361.2.c.f 4
95.i even 6 2 361.2.c.e 4
95.o odd 18 6 361.2.e.l 12
95.p even 18 6 361.2.e.k 12
285.b even 2 1 3249.2.a.n 2
380.d even 2 1 5776.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.d 2 95.d odd 2 1
361.2.a.e yes 2 5.b even 2 1
361.2.c.e 4 95.i even 6 2
361.2.c.f 4 95.h odd 6 2
361.2.e.k 12 95.p even 18 6
361.2.e.l 12 95.o odd 18 6
3249.2.a.m 2 15.d odd 2 1
3249.2.a.n 2 285.b even 2 1
5776.2.a.r 2 20.d odd 2 1
5776.2.a.bh 2 380.d even 2 1
9025.2.a.o 2 1.a even 1 1 trivial
9025.2.a.r 2 19.b odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$(T + 2)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2T - 4$$
$11$ $$T^{2} - 6T + 4$$
$13$ $$T^{2} + 3T - 9$$
$17$ $$T^{2} - T - 1$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 20$$
$29$ $$T^{2} - 3T - 29$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} + 11T + 19$$
$41$ $$T^{2} + 3T - 29$$
$43$ $$T^{2} + 8T - 4$$
$47$ $$T^{2} + 14T + 44$$
$53$ $$T^{2} + 11T - 1$$
$59$ $$T^{2} - 2T - 4$$
$61$ $$T^{2} - 13T + 31$$
$67$ $$T^{2} + 8T - 4$$
$71$ $$T^{2} + 2T - 4$$
$73$ $$T^{2} - 9T - 81$$
$79$ $$(T - 2)^{2}$$
$83$ $$T^{2} - 6T - 36$$
$89$ $$T^{2} - T - 61$$
$97$ $$T^{2} - 11T - 1$$