# Properties

 Label 9025.2.a.bf Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{6} + (2 \beta_1 - 2) q^{7} + (\beta_{3} + \beta_{2} - 2) q^{8} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q + b2 * q^2 - b3 * q^3 + (-b2 + b1 + 1) * q^4 + (2*b3 + b2 - b1 + 2) * q^6 + (2*b1 - 2) * q^7 + (b3 + b2 - 2) * q^8 + (-2*b2 + 1) * q^9 $$q + \beta_{2} q^{2} - \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{6} + (2 \beta_1 - 2) q^{7} + (\beta_{3} + \beta_{2} - 2) q^{8} + ( - 2 \beta_{2} + 1) q^{9} + 2 \beta_1 q^{11} + ( - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{12} + (\beta_{3} + 2 \beta_1) q^{13} + (2 \beta_{3} + 2 \beta_1 + 2) q^{14} + ( - 2 \beta_{3} - 2 \beta_{2} - 1) q^{16} + ( - 2 \beta_{3} - 2) q^{17} + (3 \beta_{2} - 2 \beta_1 - 6) q^{18} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{21} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{22} + (2 \beta_{3} + 2 \beta_1 + 2) q^{23} + (4 \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{24} + (\beta_{2} + 3 \beta_1) q^{26} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{28} + ( - 2 \beta_{3} - 2) q^{29} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{31} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 + 2) q^{32} + ( - 2 \beta_{2} + 2 \beta_1) q^{33} + (4 \beta_{3} - 2 \beta_1 + 4) q^{34} + ( - 2 \beta_{3} - 7 \beta_{2} + \beta_1 + 5) q^{36} + (\beta_{3} + 2 \beta_{2}) q^{37} + (2 \beta_1 - 4) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{41} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 8) q^{42} + (2 \beta_1 - 2) q^{43} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{44} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{46} + (4 \beta_{2} - 2 \beta_1 + 6) q^{47} + ( - 3 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 4) q^{48} + (4 \beta_{2} - 4 \beta_1 + 9) q^{49} + (2 \beta_{3} - 4 \beta_{2} + 8) q^{51} + (\beta_{3} + 2 \beta_{2} + 6) q^{52} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{53} + (6 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{54} + (2 \beta_{2} - 4 \beta_1 + 6) q^{56} + (4 \beta_{3} - 2 \beta_1 + 4) q^{58} + ( - 2 \beta_{2} - 2 \beta_1) q^{59} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{61} + (6 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{62} + ( - 4 \beta_{3} - 2 \beta_1 - 6) q^{63} + ( - 4 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{64} + (2 \beta_{3} + 4 \beta_{2} - 4) q^{66} + ( - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{67} + ( - 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{68} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 8) q^{69} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{71} + (5 \beta_{3} + 9 \beta_{2} - 4 \beta_1 - 4) q^{72} + (2 \beta_{3} - 6) q^{73} + ( - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 4) q^{74} + (4 \beta_{2} + 12) q^{77} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{78} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 + 1) q^{81} + (2 \beta_{3} - 2 \beta_{2} - 4) q^{82} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{83} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 12) q^{84} + (2 \beta_{3} + 2 \beta_1 + 2) q^{86} + (2 \beta_{3} - 4 \beta_{2} + 8) q^{87} + (2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 2) q^{88} + (2 \beta_{3} + 4 \beta_{2} + 2) q^{89} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 12) q^{91} + (4 \beta_{3} + 2 \beta_{2} + 10) q^{92} + ( - 8 \beta_{2} + 4 \beta_1 + 4) q^{93} + ( - 2 \beta_{3} + 2 \beta_1 + 10) q^{94} + (9 \beta_{2} - 5 \beta_1 - 6) q^{96} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{97} + ( - 4 \beta_{3} + \beta_{2} + 8) q^{98} + ( - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 4) q^{99}+O(q^{100})$$ q + b2 * q^2 - b3 * q^3 + (-b2 + b1 + 1) * q^4 + (2*b3 + b2 - b1 + 2) * q^6 + (2*b1 - 2) * q^7 + (b3 + b2 - 2) * q^8 + (-2*b2 + 1) * q^9 + 2*b1 * q^11 + (-3*b3 - 2*b2 + 2*b1 - 2) * q^12 + (b3 + 2*b1) * q^13 + (2*b3 + 2*b1 + 2) * q^14 + (-2*b3 - 2*b2 - 1) * q^16 + (-2*b3 - 2) * q^17 + (3*b2 - 2*b1 - 6) * q^18 + (2*b3 - 2*b2 + 2*b1) * q^21 + (2*b3 + 2*b2 + 2*b1 + 2) * q^22 + (2*b3 + 2*b1 + 2) * q^23 + (4*b3 + 3*b2 - b1 - 2) * q^24 + (b2 + 3*b1) * q^26 + (-2*b3 - 2*b2 + 2*b1 - 4) * q^27 + (-2*b3 + 2*b2 + 2) * q^28 + (-2*b3 - 2) * q^29 + (-2*b3 - 2*b2 + 2*b1 - 4) * q^31 + (2*b3 + b2 - 4*b1 + 2) * q^32 + (-2*b2 + 2*b1) * q^33 + (4*b3 - 2*b1 + 4) * q^34 + (-2*b3 - 7*b2 + b1 + 5) * q^36 + (b3 + 2*b2) * q^37 + (2*b1 - 4) * q^39 + (-2*b2 + 2*b1 - 6) * q^41 + (-2*b3 + 2*b2 + 2*b1 - 8) * q^42 + (2*b1 - 2) * q^43 + (-2*b3 + 2*b1 + 4) * q^44 + (-2*b3 + 2*b2 + 4*b1 - 2) * q^46 + (4*b2 - 2*b1 + 6) * q^47 + (-3*b3 - 6*b2 + 2*b1 + 4) * q^48 + (4*b2 - 4*b1 + 9) * q^49 + (2*b3 - 4*b2 + 8) * q^51 + (b3 + 2*b2 + 6) * q^52 + (-b3 - 2*b2 - 4) * q^53 + (6*b3 + 2*b2 - 2*b1) * q^54 + (2*b2 - 4*b1 + 6) * q^56 + (4*b3 - 2*b1 + 4) * q^58 + (-2*b2 - 2*b1) * q^59 + (-2*b2 + 4*b1 + 2) * q^61 + (6*b3 + 2*b2 - 2*b1) * q^62 + (-4*b3 - 2*b1 - 6) * q^63 + (-4*b3 - b2 - b1 - 3) * q^64 + (2*b3 + 4*b2 - 4) * q^66 + (-3*b3 + 2*b2 - 2*b1 - 4) * q^67 + (-6*b3 - 2*b2 + 2*b1 - 6) * q^68 + (-2*b3 + 2*b2 + 2*b1 - 8) * q^69 + (-2*b3 + 2*b2 + 2*b1 + 4) * q^71 + (5*b3 + 9*b2 - 4*b1 - 4) * q^72 + (2*b3 - 6) * q^73 + (-2*b3 - 3*b2 + 3*b1 + 4) * q^74 + (4*b2 + 12) * q^77 + (2*b3 - 2*b2 + 2*b1 + 2) * q^78 + (-2*b2 - 2*b1 + 4) * q^79 + (-2*b2 + 4*b1 + 1) * q^81 + (2*b3 - 2*b2 - 4) * q^82 + (-2*b3 + 2*b1 - 2) * q^83 + (2*b3 - 2*b2 - 2*b1 + 12) * q^84 + (2*b3 + 2*b1 + 2) * q^86 + (2*b3 - 4*b2 + 8) * q^87 + (2*b3 + 4*b2 - 4*b1 + 2) * q^88 + (2*b3 + 4*b2 + 2) * q^89 + (-2*b3 + 6*b2 - 2*b1 + 12) * q^91 + (4*b3 + 2*b2 + 10) * q^92 + (-8*b2 + 4*b1 + 4) * q^93 + (-2*b3 + 2*b1 + 10) * q^94 + (9*b2 - 5*b1 - 6) * q^96 + (-b3 - 2*b2 + 4*b1 + 4) * q^97 + (-4*b3 + b2 + 8) * q^98 + (-4*b3 - 4*b2 - 2*b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 4 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^3 + 8 * q^4 - 4 * q^7 - 12 * q^8 + 8 * q^9 $$4 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 4 q^{7} - 12 q^{8} + 8 q^{9} + 4 q^{11} + 6 q^{12} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} - 34 q^{18} + 4 q^{21} + 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26} - 4 q^{27} + 8 q^{28} - 4 q^{29} - 4 q^{31} - 6 q^{32} + 8 q^{33} + 4 q^{34} + 40 q^{36} - 6 q^{37} - 12 q^{39} - 16 q^{41} - 28 q^{42} - 4 q^{43} + 24 q^{44} + 12 q^{47} + 38 q^{48} + 20 q^{49} + 36 q^{51} + 18 q^{52} - 10 q^{53} - 20 q^{54} + 12 q^{56} + 4 q^{58} + 20 q^{61} - 20 q^{62} - 20 q^{63} - 4 q^{64} - 28 q^{66} - 18 q^{67} - 4 q^{68} - 28 q^{69} + 20 q^{71} - 52 q^{72} - 28 q^{73} + 32 q^{74} + 40 q^{77} + 12 q^{78} + 16 q^{79} + 16 q^{81} - 16 q^{82} + 44 q^{84} + 8 q^{86} + 36 q^{87} - 12 q^{88} - 4 q^{89} + 36 q^{91} + 28 q^{92} + 40 q^{93} + 48 q^{94} - 52 q^{96} + 30 q^{97} + 38 q^{98} - 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 2 * q^3 + 8 * q^4 - 4 * q^7 - 12 * q^8 + 8 * q^9 + 4 * q^11 + 6 * q^12 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 - 34 * q^18 + 4 * q^21 + 4 * q^22 + 8 * q^23 - 24 * q^24 + 4 * q^26 - 4 * q^27 + 8 * q^28 - 4 * q^29 - 4 * q^31 - 6 * q^32 + 8 * q^33 + 4 * q^34 + 40 * q^36 - 6 * q^37 - 12 * q^39 - 16 * q^41 - 28 * q^42 - 4 * q^43 + 24 * q^44 + 12 * q^47 + 38 * q^48 + 20 * q^49 + 36 * q^51 + 18 * q^52 - 10 * q^53 - 20 * q^54 + 12 * q^56 + 4 * q^58 + 20 * q^61 - 20 * q^62 - 20 * q^63 - 4 * q^64 - 28 * q^66 - 18 * q^67 - 4 * q^68 - 28 * q^69 + 20 * q^71 - 52 * q^72 - 28 * q^73 + 32 * q^74 + 40 * q^77 + 12 * q^78 + 16 * q^79 + 16 * q^81 - 16 * q^82 + 44 * q^84 + 8 * q^86 + 36 * q^87 - 12 * q^88 - 4 * q^89 + 36 * q^91 + 28 * q^92 + 40 * q^93 + 48 * q^94 - 52 * q^96 + 30 * q^97 + 38 * q^98 - 4 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 2$$ v^3 - 2*v^2 - 3*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 5\beta _1 + 4$$ b3 + 2*b2 + 5*b1 + 4

## 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
 1.28734 −0.552409 −1.51658 2.78165
−2.63010 3.04306 4.91744 0 −8.00355 0.574672 −7.67316 6.26020 0
1.2 −2.14243 −2.87834 2.59002 0 6.16666 −3.10482 −1.26409 5.28487 0
1.3 0.816594 1.53844 −1.33317 0 1.25628 −5.03316 −2.72185 −0.633188 0
1.4 1.95594 0.296842 1.82571 0 0.580605 3.56331 −0.340899 −2.91188 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.bf 4
5.b even 2 1 1805.2.a.p 4
19.b odd 2 1 475.2.a.i 4
57.d even 2 1 4275.2.a.bo 4
76.d even 2 1 7600.2.a.cf 4
95.d odd 2 1 95.2.a.b 4
95.g even 4 2 475.2.b.e 8
285.b even 2 1 855.2.a.m 4
380.d even 2 1 1520.2.a.t 4
665.g even 2 1 4655.2.a.y 4
760.b odd 2 1 6080.2.a.cc 4
760.p even 2 1 6080.2.a.ch 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 95.d odd 2 1
475.2.a.i 4 19.b odd 2 1
475.2.b.e 8 95.g even 4 2
855.2.a.m 4 285.b even 2 1
1520.2.a.t 4 380.d even 2 1
1805.2.a.p 4 5.b even 2 1
4275.2.a.bo 4 57.d even 2 1
4655.2.a.y 4 665.g even 2 1
6080.2.a.cc 4 760.b odd 2 1
6080.2.a.ch 4 760.p even 2 1
7600.2.a.cf 4 76.d even 2 1
9025.2.a.bf 4 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{4} + 2T_{2}^{3} - 6T_{2}^{2} - 8T_{2} + 9$$ T2^4 + 2*T2^3 - 6*T2^2 - 8*T2 + 9 $$T_{3}^{4} - 2T_{3}^{3} - 8T_{3}^{2} + 16T_{3} - 4$$ T3^4 - 2*T3^3 - 8*T3^2 + 16*T3 - 4 $$T_{7}^{4} + 4T_{7}^{3} - 16T_{7}^{2} - 48T_{7} + 32$$ T7^4 + 4*T7^3 - 16*T7^2 - 48*T7 + 32 $$T_{11}^{4} - 4T_{11}^{3} - 16T_{11}^{2} + 32T_{11} + 48$$ T11^4 - 4*T11^3 - 16*T11^2 + 32*T11 + 48 $$T_{29}^{4} + 4T_{29}^{3} - 32T_{29}^{2} - 16T_{29} + 48$$ T29^4 + 4*T29^3 - 32*T29^2 - 16*T29 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} - 6 T^{2} - 8 T + 9$$
$3$ $$T^{4} - 2 T^{3} - 8 T^{2} + 16 T - 4$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32$$
$11$ $$T^{4} - 4 T^{3} - 16 T^{2} + 32 T + 48$$
$13$ $$T^{4} - 2 T^{3} - 24 T^{2} + 32 T + 20$$
$17$ $$T^{4} + 4 T^{3} - 32 T^{2} - 16 T + 48$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 8 T^{3} - 24 T^{2} + 176 T + 288$$
$29$ $$T^{4} + 4 T^{3} - 32 T^{2} - 16 T + 48$$
$31$ $$T^{4} + 4 T^{3} - 80 T^{2} - 512 T - 640$$
$37$ $$T^{4} + 6 T^{3} - 24 T^{2} - 40 T + 4$$
$41$ $$T^{4} + 16 T^{3} + 56 T^{2} + \cdots - 240$$
$43$ $$T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32$$
$47$ $$T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 1056$$
$53$ $$T^{4} + 10 T^{3} - 184 T - 348$$
$59$ $$T^{4} - 64 T^{2} + 224 T - 192$$
$61$ $$T^{4} - 20 T^{3} + 56 T^{2} + \cdots - 2656$$
$67$ $$T^{4} + 18 T^{3} + 8 T^{2} + \cdots - 1076$$
$71$ $$T^{4} - 20 T^{3} + 32 T^{2} + \cdots - 4224$$
$73$ $$T^{4} + 28 T^{3} + 256 T^{2} + \cdots + 176$$
$79$ $$T^{4} - 16 T^{3} + 32 T^{2} + \cdots - 1856$$
$83$ $$T^{4} - 72 T^{2} + 112 T + 480$$
$89$ $$T^{4} + 4 T^{3} - 144 T^{2} + \cdots + 240$$
$97$ $$T^{4} - 30 T^{3} + 224 T^{2} + \cdots - 1388$$