# Properties

 Label 6525.2.a.bf 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: 3.3.469.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 4$$ x^3 - x^2 - 5*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{2} - \beta_1 + 2) q^{7} + (\beta_{2} + \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (b2 - b1 + 2) * q^7 + (b2 + b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{2} - \beta_1 + 2) q^{7} + (\beta_{2} + \beta_1) q^{8} - 3 q^{11} + ( - \beta_{2} - \beta_1 - 2) q^{13} + (3 \beta_1 - 4) q^{14} + \beta_1 q^{16} + ( - \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{2} - 2) q^{19} - 3 \beta_1 q^{22} + ( - \beta_{2} - 3 \beta_1 + 1) q^{23} + ( - 2 \beta_{2} - 3 \beta_1 - 4) q^{26} + (\beta_{2} - 2 \beta_1 + 8) q^{28} + q^{29} + ( - \beta_{2} - 2 \beta_1 + 4) q^{32} + ( - \beta_1 + 4) q^{34} + (\beta_{2} + 3 \beta_1 - 5) q^{37} + ( - 2 \beta_{2} - 4 \beta_1) q^{38} + ( - \beta_{2} + \beta_1 - 5) q^{41} + ( - \beta_{2} + 3 \beta_1 + 3) q^{43} + ( - 3 \beta_{2} - 6) q^{44} + ( - 4 \beta_{2} - 12) q^{46} + (3 \beta_{2} - \beta_1 + 2) q^{47} + (\beta_{2} - 5 \beta_1 + 5) q^{49} + ( - 3 \beta_{2} - 4 \beta_1 - 8) q^{52} + ( - 3 \beta_{2} - 3 \beta_1 - 1) q^{53} + ( - \beta_{2} + 3 \beta_1) q^{56} + \beta_1 q^{58} + (2 \beta_1 - 8) q^{59} + (2 \beta_{2} + 4) q^{61} + ( - 3 \beta_{2} + \beta_1 - 8) q^{64} + ( - \beta_{2} - 3 \beta_1 + 10) q^{67} + (\beta_{2} + 2 \beta_1 - 4) q^{68} + ( - 6 \beta_1 + 2) q^{71} + ( - 5 \beta_{2} + \beta_1 - 3) q^{73} + (4 \beta_{2} - 4 \beta_1 + 12) q^{74} + ( - 2 \beta_{2} - 2 \beta_1 - 12) q^{76} + ( - 3 \beta_{2} + 3 \beta_1 - 6) q^{77} + (4 \beta_{2} + 2 \beta_1) q^{79} + ( - 6 \beta_1 + 4) q^{82} + (\beta_{2} + \beta_1 - 5) q^{83} + (2 \beta_{2} + 2 \beta_1 + 12) q^{86} + ( - 3 \beta_{2} - 3 \beta_1) q^{88} + (3 \beta_{2} + 3 \beta_1 - 10) q^{89} + ( - \beta_{2} - \beta_1 - 4) q^{91} + ( - 2 \beta_{2} - 10 \beta_1 - 2) q^{92} + (2 \beta_{2} + 5 \beta_1 - 4) q^{94} + (\beta_{2} + 5 \beta_1 - 1) q^{97} + ( - 4 \beta_{2} + 6 \beta_1 - 20) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (b2 - b1 + 2) * q^7 + (b2 + b1) * q^8 - 3 * q^11 + (-b2 - b1 - 2) * q^13 + (3*b1 - 4) * q^14 + b1 * q^16 + (-b2 + b1) * q^17 + (-2*b2 - 2) * q^19 - 3*b1 * q^22 + (-b2 - 3*b1 + 1) * q^23 + (-2*b2 - 3*b1 - 4) * q^26 + (b2 - 2*b1 + 8) * q^28 + q^29 + (-b2 - 2*b1 + 4) * q^32 + (-b1 + 4) * q^34 + (b2 + 3*b1 - 5) * q^37 + (-2*b2 - 4*b1) * q^38 + (-b2 + b1 - 5) * q^41 + (-b2 + 3*b1 + 3) * q^43 + (-3*b2 - 6) * q^44 + (-4*b2 - 12) * q^46 + (3*b2 - b1 + 2) * q^47 + (b2 - 5*b1 + 5) * q^49 + (-3*b2 - 4*b1 - 8) * q^52 + (-3*b2 - 3*b1 - 1) * q^53 + (-b2 + 3*b1) * q^56 + b1 * q^58 + (2*b1 - 8) * q^59 + (2*b2 + 4) * q^61 + (-3*b2 + b1 - 8) * q^64 + (-b2 - 3*b1 + 10) * q^67 + (b2 + 2*b1 - 4) * q^68 + (-6*b1 + 2) * q^71 + (-5*b2 + b1 - 3) * q^73 + (4*b2 - 4*b1 + 12) * q^74 + (-2*b2 - 2*b1 - 12) * q^76 + (-3*b2 + 3*b1 - 6) * q^77 + (4*b2 + 2*b1) * q^79 + (-6*b1 + 4) * q^82 + (b2 + b1 - 5) * q^83 + (2*b2 + 2*b1 + 12) * q^86 + (-3*b2 - 3*b1) * q^88 + (3*b2 + 3*b1 - 10) * q^89 + (-b2 - b1 - 4) * q^91 + (-2*b2 - 10*b1 - 2) * q^92 + (2*b2 + 5*b1 - 4) * q^94 + (b2 + 5*b1 - 1) * q^97 + (-4*b2 + 6*b1 - 20) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 5 q^{4} + 4 q^{7}+O(q^{10})$$ 3 * q + q^2 + 5 * q^4 + 4 * q^7 $$3 q + q^{2} + 5 q^{4} + 4 q^{7} - 9 q^{11} - 6 q^{13} - 9 q^{14} + q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{22} + q^{23} - 13 q^{26} + 21 q^{28} + 3 q^{29} + 11 q^{32} + 11 q^{34} - 13 q^{37} - 2 q^{38} - 13 q^{41} + 13 q^{43} - 15 q^{44} - 32 q^{46} + 2 q^{47} + 9 q^{49} - 25 q^{52} - 3 q^{53} + 4 q^{56} + q^{58} - 22 q^{59} + 10 q^{61} - 20 q^{64} + 28 q^{67} - 11 q^{68} - 3 q^{73} + 28 q^{74} - 36 q^{76} - 12 q^{77} - 2 q^{79} + 6 q^{82} - 15 q^{83} + 36 q^{86} - 30 q^{89} - 12 q^{91} - 14 q^{92} - 9 q^{94} + q^{97} - 50 q^{98}+O(q^{100})$$ 3 * q + q^2 + 5 * q^4 + 4 * q^7 - 9 * q^11 - 6 * q^13 - 9 * q^14 + q^16 + 2 * q^17 - 4 * q^19 - 3 * q^22 + q^23 - 13 * q^26 + 21 * q^28 + 3 * q^29 + 11 * q^32 + 11 * q^34 - 13 * q^37 - 2 * q^38 - 13 * q^41 + 13 * q^43 - 15 * q^44 - 32 * q^46 + 2 * q^47 + 9 * q^49 - 25 * q^52 - 3 * q^53 + 4 * q^56 + q^58 - 22 * q^59 + 10 * q^61 - 20 * q^64 + 28 * q^67 - 11 * q^68 - 3 * q^73 + 28 * q^74 - 36 * q^76 - 12 * q^77 - 2 * q^79 + 6 * q^82 - 15 * q^83 + 36 * q^86 - 30 * q^89 - 12 * q^91 - 14 * q^92 - 9 * q^94 + q^97 - 50 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 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
 −2.16425 0.772866 2.39138
−2.16425 0 2.68397 0 0 4.84822 −1.48028 0 0
1.2 0.772866 0 −1.40268 0 0 −2.17554 −2.62981 0 0
1.3 2.39138 0 3.71871 0 0 1.32733 4.11009 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.bf 3
3.b odd 2 1 2175.2.a.u 3
5.b even 2 1 1305.2.a.q 3
15.d odd 2 1 435.2.a.i 3
15.e even 4 2 2175.2.c.m 6
60.h even 2 1 6960.2.a.cl 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.i 3 15.d odd 2 1
1305.2.a.q 3 5.b even 2 1
2175.2.a.u 3 3.b odd 2 1
2175.2.c.m 6 15.e even 4 2
6525.2.a.bf 3 1.a even 1 1 trivial
6960.2.a.cl 3 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{3} - T_{2}^{2} - 5T_{2} + 4$$ T2^3 - T2^2 - 5*T2 + 4 $$T_{7}^{3} - 4T_{7}^{2} - 7T_{7} + 14$$ T7^3 - 4*T7^2 - 7*T7 + 14 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 5T + 4$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 4 T^{2} + \cdots + 14$$
$11$ $$(T + 3)^{3}$$
$13$ $$T^{3} + 6T^{2} - T - 2$$
$17$ $$T^{3} - 2 T^{2} + \cdots + 8$$
$19$ $$T^{3} + 4 T^{2} + \cdots - 88$$
$23$ $$T^{3} - T^{2} + \cdots + 112$$
$29$ $$(T - 1)^{3}$$
$31$ $$T^{3}$$
$37$ $$T^{3} + 13T^{2} - 256$$
$41$ $$T^{3} + 13 T^{2} + \cdots + 28$$
$43$ $$T^{3} - 13 T^{2} + \cdots + 308$$
$47$ $$T^{3} - 2 T^{2} + \cdots + 266$$
$53$ $$T^{3} + 3 T^{2} + \cdots + 316$$
$59$ $$T^{3} + 22 T^{2} + \cdots + 256$$
$61$ $$T^{3} - 10 T^{2} + \cdots + 112$$
$67$ $$T^{3} - 28 T^{2} + \cdots - 194$$
$71$ $$T^{3} - 192T - 488$$
$73$ $$T^{3} + 3 T^{2} + \cdots - 1168$$
$79$ $$T^{3} + 2 T^{2} + \cdots - 224$$
$83$ $$T^{3} + 15 T^{2} + \cdots + 44$$
$89$ $$T^{3} + 30 T^{2} + \cdots - 602$$
$97$ $$T^{3} - T^{2} + \cdots - 76$$