# Properties

 Label 6525.2.a.be Level $6525$ Weight $2$ Character orbit 6525.a Self dual yes Analytic conductor $52.102$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1) * q^4 + (-b2 - b1 - 1) * q^7 + (b2 + 1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 1) q^{8} + ( - \beta_{2} + \beta_1 - 1) q^{11} + 2 \beta_1 q^{13} + ( - \beta_{2} - 3 \beta_1 - 1) q^{14} + ( - 2 \beta_{2} - 1) q^{16} + (3 \beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_{2} - \beta_1 - 3) q^{19} + (\beta_{2} - \beta_1 + 3) q^{22} + ( - \beta_{2} + \beta_1 + 5) q^{23} + (2 \beta_{2} + 2 \beta_1 + 4) q^{26} + ( - \beta_{2} - 3 \beta_1 - 3) q^{28} + q^{29} + (\beta_{2} + \beta_1 - 5) q^{31} + ( - 2 \beta_{2} - 3 \beta_1) q^{32} + ( - \beta_{2} + \beta_1 - 5) q^{34} + (3 \beta_{2} - \beta_1 + 3) q^{37} + ( - \beta_{2} - 5 \beta_1 - 1) q^{38} + (2 \beta_{2} - 4 \beta_1 + 2) q^{41} + ( - 5 \beta_{2} - 5 \beta_1 + 1) q^{43} + (\beta_{2} + \beta_1 - 1) q^{44} + (\beta_{2} + 5 \beta_1 + 3) q^{46} + ( - \beta_{2} - \beta_1 + 5) q^{47} + (2 \beta_{2} + 4 \beta_1 - 3) q^{49} + (2 \beta_{2} + 4 \beta_1 + 2) q^{52} + ( - 2 \beta_{2} + 2) q^{53} + ( - \beta_{2} - \beta_1 - 3) q^{56} + \beta_1 q^{58} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{59} - 6 \beta_1 q^{61} + (\beta_{2} - 3 \beta_1 + 1) q^{62} + (\beta_{2} - 5 \beta_1 - 2) q^{64} + (\beta_{2} - \beta_1 - 9) q^{67} + ( - 5 \beta_{2} - 3 \beta_1 + 5) q^{68} + (2 \beta_{2} - 4 \beta_1 - 8) q^{71} + (7 \beta_{2} + \beta_1 + 5) q^{73} + ( - \beta_{2} + 5 \beta_1 - 5) q^{74} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{76} + ( - 2 \beta_1 + 2) q^{77} + ( - 5 \beta_{2} - 3 \beta_1 - 1) q^{79} + ( - 4 \beta_{2} - 10) q^{82} + ( - 3 \beta_{2} - 3 \beta_1 + 5) q^{83} + ( - 5 \beta_{2} - 9 \beta_1 - 5) q^{86} + ( - \beta_{2} + 3 \beta_1 - 5) q^{88} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{89} + ( - 2 \beta_{2} - 6 \beta_1 - 2) q^{91} + (7 \beta_{2} + 7 \beta_1 - 1) q^{92} + ( - \beta_{2} + 3 \beta_1 - 1) q^{94} + ( - 3 \beta_{2} - 5 \beta_1 - 1) q^{97} + (4 \beta_{2} + 3 \beta_1 + 6) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1) * q^4 + (-b2 - b1 - 1) * q^7 + (b2 + 1) * q^8 + (-b2 + b1 - 1) * q^11 + 2*b1 * q^13 + (-b2 - 3*b1 - 1) * q^14 + (-2*b2 - 1) * q^16 + (3*b2 - b1 - 1) * q^17 + (-b2 - b1 - 3) * q^19 + (b2 - b1 + 3) * q^22 + (-b2 + b1 + 5) * q^23 + (2*b2 + 2*b1 + 4) * q^26 + (-b2 - 3*b1 - 3) * q^28 + q^29 + (b2 + b1 - 5) * q^31 + (-2*b2 - 3*b1) * q^32 + (-b2 + b1 - 5) * q^34 + (3*b2 - b1 + 3) * q^37 + (-b2 - 5*b1 - 1) * q^38 + (2*b2 - 4*b1 + 2) * q^41 + (-5*b2 - 5*b1 + 1) * q^43 + (b2 + b1 - 1) * q^44 + (b2 + 5*b1 + 3) * q^46 + (-b2 - b1 + 5) * q^47 + (2*b2 + 4*b1 - 3) * q^49 + (2*b2 + 4*b1 + 2) * q^52 + (-2*b2 + 2) * q^53 + (-b2 - b1 - 3) * q^56 + b1 * q^58 + (-2*b2 + 4*b1 - 4) * q^59 - 6*b1 * q^61 + (b2 - 3*b1 + 1) * q^62 + (b2 - 5*b1 - 2) * q^64 + (b2 - b1 - 9) * q^67 + (-5*b2 - 3*b1 + 5) * q^68 + (2*b2 - 4*b1 - 8) * q^71 + (7*b2 + b1 + 5) * q^73 + (-b2 + 5*b1 - 5) * q^74 + (-3*b2 - 5*b1 - 3) * q^76 + (-2*b1 + 2) * q^77 + (-5*b2 - 3*b1 - 1) * q^79 + (-4*b2 - 10) * q^82 + (-3*b2 - 3*b1 + 5) * q^83 + (-5*b2 - 9*b1 - 5) * q^86 + (-b2 + 3*b1 - 5) * q^88 + (-2*b2 - 2*b1 + 4) * q^89 + (-2*b2 - 6*b1 - 2) * q^91 + (7*b2 + 7*b1 - 1) * q^92 + (-b2 + 3*b1 - 1) * q^94 + (-3*b2 - 5*b1 - 1) * q^97 + (4*b2 + 3*b1 + 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + q^2 + q^4 - 4 * q^7 + 3 * q^8 $$3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8} - 2 q^{11} + 2 q^{13} - 6 q^{14} - 3 q^{16} - 4 q^{17} - 10 q^{19} + 8 q^{22} + 16 q^{23} + 14 q^{26} - 12 q^{28} + 3 q^{29} - 14 q^{31} - 3 q^{32} - 14 q^{34} + 8 q^{37} - 8 q^{38} + 2 q^{41} - 2 q^{43} - 2 q^{44} + 14 q^{46} + 14 q^{47} - 5 q^{49} + 10 q^{52} + 6 q^{53} - 10 q^{56} + q^{58} - 8 q^{59} - 6 q^{61} - 11 q^{64} - 28 q^{67} + 12 q^{68} - 28 q^{71} + 16 q^{73} - 10 q^{74} - 14 q^{76} + 4 q^{77} - 6 q^{79} - 30 q^{82} + 12 q^{83} - 24 q^{86} - 12 q^{88} + 10 q^{89} - 12 q^{91} + 4 q^{92} - 8 q^{97} + 21 q^{98}+O(q^{100})$$ 3 * q + q^2 + q^4 - 4 * q^7 + 3 * q^8 - 2 * q^11 + 2 * q^13 - 6 * q^14 - 3 * q^16 - 4 * q^17 - 10 * q^19 + 8 * q^22 + 16 * q^23 + 14 * q^26 - 12 * q^28 + 3 * q^29 - 14 * q^31 - 3 * q^32 - 14 * q^34 + 8 * q^37 - 8 * q^38 + 2 * q^41 - 2 * q^43 - 2 * q^44 + 14 * q^46 + 14 * q^47 - 5 * q^49 + 10 * q^52 + 6 * q^53 - 10 * q^56 + q^58 - 8 * q^59 - 6 * q^61 - 11 * q^64 - 28 * q^67 + 12 * q^68 - 28 * q^71 + 16 * q^73 - 10 * q^74 - 14 * q^76 + 4 * q^77 - 6 * q^79 - 30 * q^82 + 12 * q^83 - 24 * q^86 - 12 * q^88 + 10 * q^89 - 12 * q^91 + 4 * q^92 - 8 * q^97 + 21 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 −1.48119 0.311108 2.17009
−1.48119 0 0.193937 0 0 −1.19394 2.67513 0 0
1.2 0.311108 0 −1.90321 0 0 0.903212 −1.21432 0 0
1.3 2.17009 0 2.70928 0 0 −3.70928 1.53919 0 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
$$3$$ $$-1$$
$$5$$ $$+1$$
$$29$$ $$-1$$

## 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 6525.2.a.be 3
3.b odd 2 1 725.2.a.e 3
5.b even 2 1 1305.2.a.p 3
15.d odd 2 1 145.2.a.c 3
15.e even 4 2 725.2.b.e 6
60.h even 2 1 2320.2.a.n 3
105.g even 2 1 7105.2.a.o 3
120.i odd 2 1 9280.2.a.bj 3
120.m even 2 1 9280.2.a.br 3
435.b odd 2 1 4205.2.a.f 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.c 3 15.d odd 2 1
725.2.a.e 3 3.b odd 2 1
725.2.b.e 6 15.e even 4 2
1305.2.a.p 3 5.b even 2 1
2320.2.a.n 3 60.h even 2 1
4205.2.a.f 3 435.b odd 2 1
6525.2.a.be 3 1.a even 1 1 trivial
7105.2.a.o 3 105.g even 2 1
9280.2.a.bj 3 120.i odd 2 1
9280.2.a.br 3 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6525))$$:

 $$T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1$$ T2^3 - T2^2 - 3*T2 + 1 $$T_{7}^{3} + 4T_{7}^{2} - 4$$ T7^3 + 4*T7^2 - 4 $$T_{11}^{3} + 2T_{11}^{2} - 8T_{11} + 4$$ T11^3 + 2*T11^2 - 8*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 3T + 1$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 4T^{2} - 4$$
$11$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$13$ $$T^{3} - 2 T^{2} + \cdots + 8$$
$17$ $$T^{3} + 4 T^{2} + \cdots - 68$$
$19$ $$T^{3} + 10 T^{2} + \cdots + 20$$
$23$ $$T^{3} - 16 T^{2} + \cdots - 92$$
$29$ $$(T - 1)^{3}$$
$31$ $$T^{3} + 14 T^{2} + \cdots + 76$$
$37$ $$T^{3} - 8 T^{2} + \cdots + 92$$
$41$ $$T^{3} - 2 T^{2} + \cdots - 232$$
$43$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$47$ $$T^{3} - 14 T^{2} + \cdots - 76$$
$53$ $$T^{3} - 6 T^{2} + \cdots + 8$$
$59$ $$T^{3} + 8 T^{2} + \cdots + 80$$
$61$ $$T^{3} + 6 T^{2} + \cdots - 216$$
$67$ $$T^{3} + 28 T^{2} + \cdots + 716$$
$71$ $$T^{3} + 28 T^{2} + \cdots - 272$$
$73$ $$T^{3} - 16 T^{2} + \cdots + 1700$$
$79$ $$T^{3} + 6 T^{2} + \cdots - 460$$
$83$ $$T^{3} - 12T^{2} + 148$$
$89$ $$T^{3} - 10 T^{2} + \cdots + 40$$
$97$ $$T^{3} + 8 T^{2} + \cdots + 76$$