# Properties

 Label 9025.2.a.n Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $2$ 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: $$2$$ 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: 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 2) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{6} - 3 q^{7} + (2 \beta - 1) q^{8} + ( - 3 \beta + 2) q^{9} +O(q^{10})$$ q - b * q^2 + (b - 2) * q^3 + (b - 1) * q^4 + (b - 1) * q^6 - 3 * q^7 + (2*b - 1) * q^8 + (-3*b + 2) * q^9 $$q - \beta q^{2} + (\beta - 2) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{6} - 3 q^{7} + (2 \beta - 1) q^{8} + ( - 3 \beta + 2) q^{9} - \beta q^{11} + ( - 2 \beta + 3) q^{12} + q^{13} + 3 \beta q^{14} - 3 \beta q^{16} + (2 \beta - 4) q^{17} + (\beta + 3) q^{18} + ( - 3 \beta + 6) q^{21} + (\beta + 1) q^{22} + (\beta - 7) q^{23} + ( - 3 \beta + 4) q^{24} - \beta q^{26} + (2 \beta - 1) q^{27} + ( - 3 \beta + 3) q^{28} + ( - \beta - 2) q^{29} + ( - 3 \beta - 4) q^{31} + ( - \beta + 5) q^{32} + (\beta - 1) q^{33} + (2 \beta - 2) q^{34} + (2 \beta - 5) q^{36} + ( - 3 \beta - 4) q^{37} + (\beta - 2) q^{39} - 3 q^{41} + ( - 3 \beta + 3) q^{42} + ( - 3 \beta + 5) q^{43} - q^{44} + (6 \beta - 1) q^{46} - 3 q^{47} + (3 \beta - 3) q^{48} + 2 q^{49} + ( - 6 \beta + 10) q^{51} + (\beta - 1) q^{52} + (7 \beta - 5) q^{53} + ( - \beta - 2) q^{54} + ( - 6 \beta + 3) q^{56} + (3 \beta + 1) q^{58} + (7 \beta - 11) q^{59} + ( - 2 \beta - 7) q^{61} + (7 \beta + 3) q^{62} + (9 \beta - 6) q^{63} + (2 \beta + 1) q^{64} - q^{66} + 7 q^{67} + ( - 4 \beta + 6) q^{68} + ( - 8 \beta + 15) q^{69} + ( - 4 \beta - 1) q^{71} + (\beta - 8) q^{72} + (6 \beta - 7) q^{73} + (7 \beta + 3) q^{74} + 3 \beta q^{77} + (\beta - 1) q^{78} + (12 \beta - 6) q^{79} + (6 \beta - 2) q^{81} + 3 \beta q^{82} + ( - 4 \beta - 2) q^{83} + (6 \beta - 9) q^{84} + ( - 2 \beta + 3) q^{86} + ( - \beta + 3) q^{87} + ( - \beta - 2) q^{88} + (2 \beta - 11) q^{89} - 3 q^{91} + ( - 7 \beta + 8) q^{92} + ( - \beta + 5) q^{93} + 3 \beta q^{94} + (6 \beta - 11) q^{96} + ( - 3 \beta - 9) q^{97} - 2 \beta q^{98} + (\beta + 3) q^{99} +O(q^{100})$$ q - b * q^2 + (b - 2) * q^3 + (b - 1) * q^4 + (b - 1) * q^6 - 3 * q^7 + (2*b - 1) * q^8 + (-3*b + 2) * q^9 - b * q^11 + (-2*b + 3) * q^12 + q^13 + 3*b * q^14 - 3*b * q^16 + (2*b - 4) * q^17 + (b + 3) * q^18 + (-3*b + 6) * q^21 + (b + 1) * q^22 + (b - 7) * q^23 + (-3*b + 4) * q^24 - b * q^26 + (2*b - 1) * q^27 + (-3*b + 3) * q^28 + (-b - 2) * q^29 + (-3*b - 4) * q^31 + (-b + 5) * q^32 + (b - 1) * q^33 + (2*b - 2) * q^34 + (2*b - 5) * q^36 + (-3*b - 4) * q^37 + (b - 2) * q^39 - 3 * q^41 + (-3*b + 3) * q^42 + (-3*b + 5) * q^43 - q^44 + (6*b - 1) * q^46 - 3 * q^47 + (3*b - 3) * q^48 + 2 * q^49 + (-6*b + 10) * q^51 + (b - 1) * q^52 + (7*b - 5) * q^53 + (-b - 2) * q^54 + (-6*b + 3) * q^56 + (3*b + 1) * q^58 + (7*b - 11) * q^59 + (-2*b - 7) * q^61 + (7*b + 3) * q^62 + (9*b - 6) * q^63 + (2*b + 1) * q^64 - q^66 + 7 * q^67 + (-4*b + 6) * q^68 + (-8*b + 15) * q^69 + (-4*b - 1) * q^71 + (b - 8) * q^72 + (6*b - 7) * q^73 + (7*b + 3) * q^74 + 3*b * q^77 + (b - 1) * q^78 + (12*b - 6) * q^79 + (6*b - 2) * q^81 + 3*b * q^82 + (-4*b - 2) * q^83 + (6*b - 9) * q^84 + (-2*b + 3) * q^86 + (-b + 3) * q^87 + (-b - 2) * q^88 + (2*b - 11) * q^89 - 3 * q^91 + (-7*b + 8) * q^92 + (-b + 5) * q^93 + 3*b * q^94 + (6*b - 11) * q^96 + (-3*b - 9) * q^97 - 2*b * q^98 + (b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9}+O(q^{10})$$ 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9 $$2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9} - q^{11} + 4 q^{12} + 2 q^{13} + 3 q^{14} - 3 q^{16} - 6 q^{17} + 7 q^{18} + 9 q^{21} + 3 q^{22} - 13 q^{23} + 5 q^{24} - q^{26} + 3 q^{28} - 5 q^{29} - 11 q^{31} + 9 q^{32} - q^{33} - 2 q^{34} - 8 q^{36} - 11 q^{37} - 3 q^{39} - 6 q^{41} + 3 q^{42} + 7 q^{43} - 2 q^{44} + 4 q^{46} - 6 q^{47} - 3 q^{48} + 4 q^{49} + 14 q^{51} - q^{52} - 3 q^{53} - 5 q^{54} + 5 q^{58} - 15 q^{59} - 16 q^{61} + 13 q^{62} - 3 q^{63} + 4 q^{64} - 2 q^{66} + 14 q^{67} + 8 q^{68} + 22 q^{69} - 6 q^{71} - 15 q^{72} - 8 q^{73} + 13 q^{74} + 3 q^{77} - q^{78} + 2 q^{81} + 3 q^{82} - 8 q^{83} - 12 q^{84} + 4 q^{86} + 5 q^{87} - 5 q^{88} - 20 q^{89} - 6 q^{91} + 9 q^{92} + 9 q^{93} + 3 q^{94} - 16 q^{96} - 21 q^{97} - 2 q^{98} + 7 q^{99}+O(q^{100})$$ 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9 - q^11 + 4 * q^12 + 2 * q^13 + 3 * q^14 - 3 * q^16 - 6 * q^17 + 7 * q^18 + 9 * q^21 + 3 * q^22 - 13 * q^23 + 5 * q^24 - q^26 + 3 * q^28 - 5 * q^29 - 11 * q^31 + 9 * q^32 - q^33 - 2 * q^34 - 8 * q^36 - 11 * q^37 - 3 * q^39 - 6 * q^41 + 3 * q^42 + 7 * q^43 - 2 * q^44 + 4 * q^46 - 6 * q^47 - 3 * q^48 + 4 * q^49 + 14 * q^51 - q^52 - 3 * q^53 - 5 * q^54 + 5 * q^58 - 15 * q^59 - 16 * q^61 + 13 * q^62 - 3 * q^63 + 4 * q^64 - 2 * q^66 + 14 * q^67 + 8 * q^68 + 22 * q^69 - 6 * q^71 - 15 * q^72 - 8 * q^73 + 13 * q^74 + 3 * q^77 - q^78 + 2 * q^81 + 3 * q^82 - 8 * q^83 - 12 * q^84 + 4 * q^86 + 5 * q^87 - 5 * q^88 - 20 * q^89 - 6 * q^91 + 9 * q^92 + 9 * q^93 + 3 * q^94 - 16 * q^96 - 21 * q^97 - 2 * q^98 + 7 * 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
−1.61803 −0.381966 0.618034 0 0.618034 −3.00000 2.23607 −2.85410 0
1.2 0.618034 −2.61803 −1.61803 0 −1.61803 −3.00000 −2.23607 3.85410 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.n 2
5.b even 2 1 361.2.a.f yes 2
15.d odd 2 1 3249.2.a.i 2
19.b odd 2 1 9025.2.a.s 2
20.d odd 2 1 5776.2.a.s 2
95.d odd 2 1 361.2.a.c 2
95.h odd 6 2 361.2.c.g 4
95.i even 6 2 361.2.c.d 4
95.o odd 18 6 361.2.e.i 12
95.p even 18 6 361.2.e.j 12
285.b even 2 1 3249.2.a.o 2
380.d even 2 1 5776.2.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.c 2 95.d odd 2 1
361.2.a.f yes 2 5.b even 2 1
361.2.c.d 4 95.i even 6 2
361.2.c.g 4 95.h odd 6 2
361.2.e.i 12 95.o odd 18 6
361.2.e.j 12 95.p even 18 6
3249.2.a.i 2 15.d odd 2 1
3249.2.a.o 2 285.b even 2 1
5776.2.a.s 2 20.d odd 2 1
5776.2.a.bg 2 380.d even 2 1
9025.2.a.n 2 1.a even 1 1 trivial
9025.2.a.s 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2} + 3T + 1$$
$5$ $$T^{2}$$
$7$ $$(T + 3)^{2}$$
$11$ $$T^{2} + T - 1$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 6T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 13T + 41$$
$29$ $$T^{2} + 5T + 5$$
$31$ $$T^{2} + 11T + 19$$
$37$ $$T^{2} + 11T + 19$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} - 7T + 1$$
$47$ $$(T + 3)^{2}$$
$53$ $$T^{2} + 3T - 59$$
$59$ $$T^{2} + 15T - 5$$
$61$ $$T^{2} + 16T + 59$$
$67$ $$(T - 7)^{2}$$
$71$ $$T^{2} + 6T - 11$$
$73$ $$T^{2} + 8T - 29$$
$79$ $$T^{2} - 180$$
$83$ $$T^{2} + 8T - 4$$
$89$ $$T^{2} + 20T + 95$$
$97$ $$T^{2} + 21T + 99$$