# Properties

 Label 6525.2.a.bs Level $6525$ Weight $2$ Character orbit 6525.a Self dual yes Analytic conductor $52.102$ Analytic rank $1$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.246832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ x^5 - 2*x^4 - 5*x^3 + 6*x^2 + 7*x - 2 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 1) q^{4} + (\beta_{2} - 2) q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 - b1 + 1) * q^4 + (b2 - 2) * q^7 + (-b3 + b2 - b1 + 2) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 1) q^{4} + (\beta_{2} - 2) q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{8} + (\beta_{3} + \beta_1 - 3) q^{11} - \beta_{2} q^{13} + ( - \beta_{3} - 1) q^{14} + (\beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{16} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{17} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \cdots + 1) q^{19}+ \cdots + (\beta_{4} + 3 \beta_{3} - \beta_{2} + \cdots - 5) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 - b1 + 1) * q^4 + (b2 - 2) * q^7 + (-b3 + b2 - b1 + 2) * q^8 + (b3 + b1 - 3) * q^11 - b2 * q^13 + (-b3 - 1) * q^14 + (b4 - b3 - 2*b1 + 1) * q^16 + (-b4 - 2*b2 + 1) * q^17 + (-2*b4 - b3 - 3*b2 + b1 + 1) * q^19 + (-b4 - 2*b2 + 3*b1 - 3) * q^22 + (b4 + b3 + 2*b2 + b1) * q^23 + (b3 + 2*b1 - 1) * q^26 + (b4 - b2 + b1 + 1) * q^28 - q^29 + (3*b4 + 3*b3 + 2*b2 + b1 - 2) * q^31 + (3*b4 + 2*b3 + 2*b2 + b1 - 2) * q^32 + (-2*b4 + 2*b3 - b2 + 3*b1) * q^34 + (b4 - b3 - 2*b2 - b1 - 2) * q^37 + (-3*b4 + 3*b3 - 2*b2 + 5*b1 - 4) * q^38 + (2*b4 - 2*b3 + 2*b2 + 2) * q^41 + (b4 + 4*b3 + b2 + 4*b1 - 3) * q^43 + (-2*b4 - 4*b2 + 5*b1 - 4) * q^44 + (b4 - 2*b3 - b2 - 4*b1 + 1) * q^46 + (-b4 + 4*b1 - 1) * q^47 + (b4 + b3 - 3*b2 + b1 - 1) * q^49 + (-b4 - b2 + b1 - 3) * q^52 + (2*b4 - b3 - b2 + b1 + 5) * q^53 + (2*b4 + 3*b3 + b1 - 1) * q^56 + (b1 - 1) * q^58 + (-5*b4 - 2*b3 - b2 + 1) * q^59 + (-2*b4 - b3 + b2 - 3*b1 - 1) * q^61 + (3*b4 - 2*b3 - b2 - 2*b1 + 1) * q^62 + (2*b4 + 2*b1 - 3) * q^64 + (-2*b4 + 2*b3 - b2 - 2) * q^67 + (-4*b4 + b3 - 3*b2 + 2*b1 - 3) * q^68 + (2*b4 - 3*b3 + b2 - 3*b1 - 5) * q^71 + (-b4 - 5*b3 - b1 + 4) * q^73 + (3*b4 + 2*b3 + 3*b2 + 6*b1 - 5) * q^74 + (-5*b4 + 4*b3 - 5*b2 + 6*b1 - 9) * q^76 + (-b4 - 2*b3 - 2*b2 + 5) * q^77 + (-5*b3 - b2 - 3*b1 - 1) * q^79 + (6*b4 - 2*b3 + 4*b2 - 6*b1 - 2) * q^82 + (-b4 + 2*b3 - b2 + 4*b1 - 1) * q^83 + (-2*b4 - b3 - 7*b2 + b1 - 3) * q^86 + (-2*b4 + 4*b3 - 3*b2 + 6*b1 - 10) * q^88 + (4*b4 - b3 - 3*b1 + 5) * q^89 + (-b4 - b3 + b2 - b1 - 2) * q^91 + (2*b4 - b3 + 3*b2 - b1 + 3) * q^92 + (-2*b4 - 5*b2 + b1 - 8) * q^94 + (-b3 - b2 - b1 - 3) * q^97 + (b4 + 3*b3 - b2 + 7*b1 - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8}+O(q^{10})$$ 5 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8 $$5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} - 12 q^{11} - 2 q^{13} - 6 q^{14} + q^{16} - 2 q^{19} - 14 q^{22} + 8 q^{23} + 6 q^{28} - 5 q^{29} + 2 q^{31} + q^{32} + 4 q^{34} - 16 q^{37} - 14 q^{38} + 14 q^{41} - 20 q^{44} - 6 q^{46} + 2 q^{47} - 7 q^{49} - 16 q^{52} + 26 q^{53} + 2 q^{56} - 3 q^{58} - 4 q^{59} - 12 q^{61} - 9 q^{64} - 12 q^{67} - 20 q^{68} - 30 q^{71} + 12 q^{73} - 2 q^{74} - 44 q^{76} + 18 q^{77} - 18 q^{79} - 10 q^{82} + 2 q^{83} - 30 q^{86} - 42 q^{88} + 22 q^{89} - 12 q^{91} + 20 q^{92} - 50 q^{94} - 20 q^{97} - 9 q^{98}+O(q^{100})$$ 5 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8 - 12 * q^11 - 2 * q^13 - 6 * q^14 + q^16 - 2 * q^19 - 14 * q^22 + 8 * q^23 + 6 * q^28 - 5 * q^29 + 2 * q^31 + q^32 + 4 * q^34 - 16 * q^37 - 14 * q^38 + 14 * q^41 - 20 * q^44 - 6 * q^46 + 2 * q^47 - 7 * q^49 - 16 * q^52 + 26 * q^53 + 2 * q^56 - 3 * q^58 - 4 * q^59 - 12 * q^61 - 9 * q^64 - 12 * q^67 - 20 * q^68 - 30 * q^71 + 12 * q^73 - 2 * q^74 - 44 * q^76 + 18 * q^77 - 18 * q^79 - 10 * q^82 + 2 * q^83 - 30 * q^86 - 42 * q^88 + 22 * q^89 - 12 * q^91 + 20 * q^92 - 50 * q^94 - 20 * q^97 - 9 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 3$$ v^3 - 2*v^2 - 3*v + 3 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1$$ v^4 - 3*v^3 - 2*v^2 + 7*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 5\beta _1 + 1$$ b3 + 2*b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3\beta_{3} + 8\beta_{2} + 10\beta _1 + 6$$ b4 + 3*b3 + 8*b2 + 10*b1 + 6

## 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
 2.71457 1.71250 0.245526 −1.15351 −1.51908
−1.71457 0 0.939748 0 0 0.654317 1.81788 0 0
1.2 −0.712495 0 −1.49235 0 0 −2.77986 2.48828 0 0
1.3 0.754474 0 −1.43077 0 0 −4.18524 −2.58843 0 0
1.4 2.15351 0 2.63760 0 0 −1.51591 1.37308 0 0
1.5 2.51908 0 4.34577 0 0 −0.173311 5.90919 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.bs 5
3.b odd 2 1 2175.2.a.w 5
5.b even 2 1 6525.2.a.bl 5
5.c odd 4 2 1305.2.c.j 10
15.d odd 2 1 2175.2.a.z 5
15.e even 4 2 435.2.c.e 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.e 10 15.e even 4 2
1305.2.c.j 10 5.c odd 4 2
2175.2.a.w 5 3.b odd 2 1
2175.2.a.z 5 15.d odd 2 1
6525.2.a.bl 5 5.b even 2 1
6525.2.a.bs 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6525))$$:

 $$T_{2}^{5} - 3T_{2}^{4} - 3T_{2}^{3} + 11T_{2}^{2} + T_{2} - 5$$ T2^5 - 3*T2^4 - 3*T2^3 + 11*T2^2 + T2 - 5 $$T_{7}^{5} + 8T_{7}^{4} + 18T_{7}^{3} + 6T_{7}^{2} - 11T_{7} - 2$$ T7^5 + 8*T7^4 + 18*T7^3 + 6*T7^2 - 11*T7 - 2 $$T_{11}^{5} + 12T_{11}^{4} + 48T_{11}^{3} + 72T_{11}^{2} + 35T_{11} + 4$$ T11^5 + 12*T11^4 + 48*T11^3 + 72*T11^2 + 35*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 3 T^{4} + \cdots - 5$$
$3$ $$T^{5}$$
$5$ $$T^{5}$$
$7$ $$T^{5} + 8 T^{4} + \cdots - 2$$
$11$ $$T^{5} + 12 T^{4} + \cdots + 4$$
$13$ $$T^{5} + 2 T^{4} + \cdots + 4$$
$17$ $$T^{5} - 28 T^{3} + \cdots - 298$$
$19$ $$T^{5} + 2 T^{4} + \cdots - 304$$
$23$ $$T^{5} - 8 T^{4} + \cdots - 40$$
$29$ $$(T + 1)^{5}$$
$31$ $$T^{5} - 2 T^{4} + \cdots - 6304$$
$37$ $$T^{5} + 16 T^{4} + \cdots + 584$$
$41$ $$T^{5} - 14 T^{4} + \cdots - 6176$$
$43$ $$T^{5} - 146 T^{3} + \cdots + 6848$$
$47$ $$T^{5} - 2 T^{4} + \cdots + 2692$$
$53$ $$T^{5} - 26 T^{4} + \cdots + 15056$$
$59$ $$T^{5} + 4 T^{4} + \cdots - 2000$$
$61$ $$T^{5} + 12 T^{4} + \cdots + 6872$$
$67$ $$T^{5} + 12 T^{4} + \cdots - 1310$$
$71$ $$T^{5} + 30 T^{4} + \cdots - 6592$$
$73$ $$T^{5} - 12 T^{4} + \cdots - 3368$$
$79$ $$T^{5} + 18 T^{4} + \cdots + 52048$$
$83$ $$T^{5} - 2 T^{4} + \cdots + 8$$
$89$ $$T^{5} - 22 T^{4} + \cdots - 40682$$
$97$ $$T^{5} + 20 T^{4} + \cdots + 328$$