# Properties

 Label 9025.2.a.bg Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.7537.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 4x + 3$$ x^4 - x^3 - 5*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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_1 - 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{4} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} + (\beta_{3} + 1) q^{7} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8} + (\beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b1 - 1) * q^3 + (b3 - b2 + 1) * q^4 + (b3 - b2 + b1) * q^6 + (b3 + 1) * q^7 + (-2*b3 + b2 + b1 - 3) * q^8 + (b2 - 2*b1 + 1) * q^9 $$q + \beta_{2} q^{2} + (\beta_1 - 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{4} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} + (\beta_{3} + 1) q^{7} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8} + (\beta_{2} - 2 \beta_1 + 1) q^{9} + ( - \beta_{3} - 2 \beta_1) q^{11} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{12} + (\beta_{3} + \beta_1 - 2) q^{13} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{14} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{16} + (\beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{3} - 2 \beta_1 + 3) q^{18} + (\beta_{2} + \beta_1 - 1) q^{21} + ( - \beta_{3} - \beta_{2} - 3 \beta_1) q^{22} + (\beta_{3} + \beta_{2} - \beta_1) q^{23} + (\beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{24} + ( - \beta_{2} + 2 \beta_1) q^{26} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{27} + (2 \beta_{3} - 3 \beta_{2} + 4) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{31} + ( - 3 \beta_{3} + 5 \beta_{2} + \cdots - 6) q^{32}+ \cdots + (5 \beta_{2} - 5 \beta_1 + 12) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b1 - 1) * q^3 + (b3 - b2 + 1) * q^4 + (b3 - b2 + b1) * q^6 + (b3 + 1) * q^7 + (-2*b3 + b2 + b1 - 3) * q^8 + (b2 - 2*b1 + 1) * q^9 + (-b3 - 2*b1) * q^11 + (-b3 + 2*b2 - 1) * q^12 + (b3 + b1 - 2) * q^13 + (-b3 + 2*b2 + b1) * q^14 + (2*b3 - 4*b2 - b1 + 1) * q^16 + (b2 + 2*b1) * q^17 + (-b3 - 2*b1 + 3) * q^18 + (b2 + b1 - 1) * q^21 + (-b3 - b2 - 3*b1) * q^22 + (b3 + b2 - b1) * q^23 + (b3 - 2*b2 - 3*b1 + 6) * q^24 + (-b2 + 2*b1) * q^26 + (b3 - 3*b2 + b1 - 4) * q^27 + (2*b3 - 3*b2 + 4) * q^28 + (-2*b3 + 2*b2 + b1) * q^29 + (-b3 - 3*b2 + b1 - 1) * q^31 + (-3*b3 + 5*b2 - b1 - 6) * q^32 + (-3*b2 + 2*b1 - 6) * q^33 + (3*b3 - b2 + 2*b1 + 3) * q^34 + (-b3 + b1 - 2) * q^36 + (-b3 + 2*b2 + 1) * q^37 + (2*b2 - 3*b1 + 5) * q^39 + (-3*b3 - 2*b2 + 2*b1 - 3) * q^41 + (2*b3 - 2*b2 + b1 + 3) * q^42 + (b3 + b2 - 4*b1 + 1) * q^43 + (-b3 - 3) * q^44 + (-b3 + 3) * q^46 + (-3*b3 - 2*b2 - 2*b1 + 3) * q^47 + (-4*b3 + 5*b2 - 2*b1 - 4) * q^48 + (b3 - b2 + b1 - 3) * q^49 + (b3 + b2 - b1 + 6) * q^51 + (-b3 + b2 + 1) * q^52 + (2*b3 - 3*b2 + 2*b1) * q^53 + (-3*b3 + 2*b1 - 9) * q^54 + (-3*b3 + 5*b2 - 9) * q^56 + (5*b3 - 4*b2 - b1 + 6) * q^58 - 5*b1 * q^59 + (-2*b3 - 4*b2 - 1) * q^61 + (-b3 + b2 - 9) * q^62 + (-2*b3 - b1 + 1) * q^63 + (3*b3 - 6*b2 - 2*b1 + 13) * q^64 + (-b3 - 3*b2 + 2*b1 - 9) * q^66 + (-2*b2 + 2*b1 - 2) * q^67 + (-2*b3 + 5*b2 + b1 - 3) * q^68 + (b3 - b2 + 2*b1 - 3) * q^69 + (3*b2 - b1 + 6) * q^71 + (4*b3 - 3*b2 + 4*b1 - 6) * q^72 + (b3 - b2 + 3*b1 + 4) * q^73 + (3*b3 - 2*b2 - b1 + 6) * q^74 + (-2*b3 - b2 - 3*b1 - 3) * q^77 + (-b3 + 3*b2 - 3*b1 + 6) * q^78 + (b3 + 2*b2 - b1 + 5) * q^79 + (-3*b3 + 2*b2 - 2*b1 + 4) * q^81 + (3*b3 - 4*b2 - b1 - 6) * q^82 + (2*b3 - b2 - 2*b1) * q^83 + (-3*b3 + 5*b2 + b1 - 4) * q^84 + (-4*b3 + b2 - 3*b1 + 3) * q^86 + (2*b3 - 3*b2 + b1 + 3) * q^87 + (3*b3 - 2*b2 + 5*b1) * q^88 + (-3*b3 - 2*b2 - 3*b1 + 3) * q^89 + (-b3 + 2*b1 + 1) * q^91 + (-b3 + b1) * q^92 + (-3*b3 + 3*b2 - 5*b1 + 4) * q^93 + (-b3 + 2*b2 - 5*b1 - 6) * q^94 + (5*b3 - 9*b2 + 3) * q^96 + (4*b3 - 5*b1 + 1) * q^97 + (-b3 - b2 + 2*b1 - 3) * q^98 + (5*b2 - 5*b1 + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} - 12 q^{8} + q^{9}+O(q^{10})$$ 4 * q - q^2 - 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 - 12 * q^8 + q^9 $$4 q - q^{2} - 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} - 12 q^{8} + q^{9} - 2 q^{11} - 6 q^{12} - 7 q^{13} - q^{14} + 7 q^{16} + q^{17} + 10 q^{18} - 4 q^{21} - 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26} - 12 q^{27} + 19 q^{28} - q^{29} - 30 q^{32} - 19 q^{33} + 15 q^{34} - 7 q^{36} + 2 q^{37} + 15 q^{39} - 8 q^{41} + 15 q^{42} - q^{43} - 12 q^{44} + 12 q^{46} + 12 q^{47} - 23 q^{48} - 10 q^{49} + 22 q^{51} + 3 q^{52} + 5 q^{53} - 34 q^{54} - 41 q^{56} + 27 q^{58} - 5 q^{59} - 37 q^{62} + 3 q^{63} + 56 q^{64} - 31 q^{66} - 4 q^{67} - 16 q^{68} - 9 q^{69} + 20 q^{71} - 17 q^{72} + 20 q^{73} + 25 q^{74} - 14 q^{77} + 18 q^{78} + 17 q^{79} + 12 q^{81} - 21 q^{82} - q^{83} - 20 q^{84} + 8 q^{86} + 16 q^{87} + 7 q^{88} + 11 q^{89} + 6 q^{91} + q^{92} + 8 q^{93} - 31 q^{94} + 21 q^{96} - q^{97} - 9 q^{98} + 38 q^{99}+O(q^{100})$$ 4 * q - q^2 - 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 - 12 * q^8 + q^9 - 2 * q^11 - 6 * q^12 - 7 * q^13 - q^14 + 7 * q^16 + q^17 + 10 * q^18 - 4 * q^21 - 2 * q^22 - 2 * q^23 + 23 * q^24 + 3 * q^26 - 12 * q^27 + 19 * q^28 - q^29 - 30 * q^32 - 19 * q^33 + 15 * q^34 - 7 * q^36 + 2 * q^37 + 15 * q^39 - 8 * q^41 + 15 * q^42 - q^43 - 12 * q^44 + 12 * q^46 + 12 * q^47 - 23 * q^48 - 10 * q^49 + 22 * q^51 + 3 * q^52 + 5 * q^53 - 34 * q^54 - 41 * q^56 + 27 * q^58 - 5 * q^59 - 37 * q^62 + 3 * q^63 + 56 * q^64 - 31 * q^66 - 4 * q^67 - 16 * q^68 - 9 * q^69 + 20 * q^71 - 17 * q^72 + 20 * q^73 + 25 * q^74 - 14 * q^77 + 18 * q^78 + 17 * q^79 + 12 * q^81 - 21 * q^82 - q^83 - 20 * q^84 + 8 * q^86 + 16 * q^87 + 7 * q^88 + 11 * q^89 + 6 * q^91 + q^92 + 8 * q^93 - 31 * q^94 + 21 * q^96 - q^97 - 9 * q^98 + 38 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 5x^{2} + 4x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*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
 −0.491918 1.37933 −2.04717 2.15976
−2.75802 −1.49192 5.60665 0 4.11474 2.84864 −9.94721 −0.774179 0
1.2 −1.09744 0.379334 −0.795629 0 −0.416295 −1.89307 3.06803 −2.85611 0
1.3 1.19091 −3.04717 −0.581734 0 −3.62891 0.609175 −3.07461 6.28525 0
1.4 1.66454 1.15976 0.770710 0 1.93047 2.43525 −2.04621 −1.65497 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.bg 4
5.b even 2 1 1805.2.a.o 4
19.b odd 2 1 9025.2.a.bp 4
19.c even 3 2 475.2.e.e 8
95.d odd 2 1 1805.2.a.i 4
95.i even 6 2 95.2.e.c 8
95.m odd 12 4 475.2.j.c 16
285.n odd 6 2 855.2.k.h 8
380.p odd 6 2 1520.2.q.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 95.i even 6 2
475.2.e.e 8 19.c even 3 2
475.2.j.c 16 95.m odd 12 4
855.2.k.h 8 285.n odd 6 2
1520.2.q.o 8 380.p odd 6 2
1805.2.a.i 4 95.d odd 2 1
1805.2.a.o 4 5.b even 2 1
9025.2.a.bg 4 1.a even 1 1 trivial
9025.2.a.bp 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2}^{4} + T_{2}^{3} - 6T_{2}^{2} - T_{2} + 6$$ T2^4 + T2^3 - 6*T2^2 - T2 + 6 $$T_{3}^{4} + 3T_{3}^{3} - 2T_{3}^{2} - 5T_{3} + 2$$ T3^4 + 3*T3^3 - 2*T3^2 - 5*T3 + 2 $$T_{7}^{4} - 4T_{7}^{3} - T_{7}^{2} + 15T_{7} - 8$$ T7^4 - 4*T7^3 - T7^2 + 15*T7 - 8 $$T_{11}^{4} + 2T_{11}^{3} - 25T_{11}^{2} - 19T_{11} + 3$$ T11^4 + 2*T11^3 - 25*T11^2 - 19*T11 + 3 $$T_{29}^{4} + T_{29}^{3} - 63T_{29}^{2} + 194T_{29} - 141$$ T29^4 + T29^3 - 63*T29^2 + 194*T29 - 141

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} - 6 T^{2} + \cdots + 6$$
$3$ $$T^{4} + 3 T^{3} + \cdots + 2$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 4 T^{3} + \cdots - 8$$
$11$ $$T^{4} + 2 T^{3} + \cdots + 3$$
$13$ $$T^{4} + 7 T^{3} + \cdots - 16$$
$17$ $$T^{4} - T^{3} + \cdots + 108$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 2 T^{3} + \cdots + 6$$
$29$ $$T^{4} + T^{3} + \cdots - 141$$
$31$ $$T^{4} - 67 T^{2} + \cdots + 1063$$
$37$ $$T^{4} - 2 T^{3} + \cdots - 118$$
$41$ $$T^{4} + 8 T^{3} + \cdots - 2238$$
$43$ $$T^{4} + T^{3} + \cdots + 794$$
$47$ $$T^{4} - 12 T^{3} + \cdots - 2316$$
$53$ $$T^{4} - 5 T^{3} + \cdots - 54$$
$59$ $$T^{4} + 5 T^{3} + \cdots + 1875$$
$61$ $$T^{4} - 130 T^{2} + \cdots + 3049$$
$67$ $$T^{4} + 4 T^{3} + \cdots + 64$$
$71$ $$T^{4} - 20 T^{3} + \cdots - 243$$
$73$ $$T^{4} - 20 T^{3} + \cdots - 1726$$
$79$ $$T^{4} - 17 T^{3} + \cdots - 184$$
$83$ $$T^{4} + T^{3} + \cdots + 366$$
$89$ $$T^{4} - 11 T^{3} + \cdots - 3816$$
$97$ $$T^{4} + T^{3} + \cdots + 7442$$