# Properties

 Label 6525.2.a.bi Level $6525$ Weight $2$ Character orbit 6525.a Self dual yes Analytic conductor $52.102$ 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] = [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: $$4$$ Coefficient field: 4.4.2225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{3} + \beta_{2}) q^{7} + (\beta_{3} - 2 \beta_{2} - 4) q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 - b1 + 2) * q^4 + (b3 + b2) * q^7 + (b3 - 2*b2 - 4) * q^8 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{3} + \beta_{2}) q^{7} + (\beta_{3} - 2 \beta_{2} - 4) q^{8} + ( - \beta_{3} - \beta_{2}) q^{11} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{13} + (\beta_{2} + \beta_1 + 1) q^{14} + ( - 3 \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{16}+ \cdots + (\beta_{2} - 4 \beta_1 + 10) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 - b1 + 2) * q^4 + (b3 + b2) * q^7 + (b3 - 2*b2 - 4) * q^8 + (-b3 - b2) * q^11 + (-b3 + b2 + 2*b1 + 2) * q^13 + (b2 + b1 + 1) * q^14 + (-3*b3 + 2*b2 - b1 + 4) * q^16 + (-b3 + b2 - 2) * q^17 + (2*b3 - 2*b1) * q^19 + (-b2 - b1 - 1) * q^22 + (-2*b3 + 2*b2 - 2) * q^23 + (2*b3 - b2 + b1 + 1) * q^26 + (-b3 - 2*b2 + b1 + 1) * q^28 + q^29 + (2*b3 - 2*b2 - 2) * q^31 + (3*b3 - 5*b2 + b1 - 7) * q^32 + (2*b3 - 3*b2 - 3*b1 - 1) * q^34 + (-2*b2 - 4*b1 + 4) * q^37 + (-2*b3 + 2*b2 + 2*b1 - 2) * q^38 + (-4*b2 - 4*b1 + 2) * q^41 + (2*b3 - 4*b2 - 2*b1 - 2) * q^43 + (b3 + 2*b2 - b1 - 1) * q^44 + (4*b3 - 6*b2 - 4*b1 - 4) * q^46 + (b3 - b2 + 2*b1 - 4) * q^47 + (-3*b3 - 3*b2 + 4*b1 - 1) * q^49 + (-b3 + 4*b2 - b1 + 3) * q^52 + (2*b3 - 4*b2 - 2*b1 - 4) * q^53 + (-b3 - b2 - 2*b1) * q^56 + (b1 - 1) * q^58 + (6*b2 + 2*b1 + 2) * q^59 + (-4*b3 - 2*b2 + 2*b1 - 8) * q^61 + (-4*b3 + 6*b2 + 8) * q^62 + (-2*b3 + 8*b2 - 2*b1 + 13) * q^64 + (-b3 - b2 - 4*b1) * q^67 + (-3*b3 + 2*b2 + b1 + 3) * q^68 + (2*b3 + 2*b1 + 2) * q^71 + (2*b2 + 4*b1) * q^73 + (-2*b3 - 2*b2 + 4*b1 - 14) * q^74 + (-4*b2 + 2) * q^76 + (3*b3 + 3*b2 - 4*b1 - 6) * q^77 + (-2*b3 + 6*b1 + 4) * q^79 + (-4*b3 + 2*b1 - 10) * q^82 + (2*b3 + 4*b2 - 2*b1) * q^83 + (-6*b3 + 6*b2 + 4) * q^86 + (b3 + b2 + 2*b1) * q^88 + (-b3 - 3*b2 + 6*b1 - 2) * q^89 + (3*b3 + 5*b2 + 2*b1) * q^91 + (-6*b3 + 6*b2 + 10) * q^92 + (-2*b3 + 5*b2 - 3*b1 + 13) * q^94 + (-4*b3 - 2*b1 + 6) * q^97 + (b2 - 4*b1 + 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} + 5 q^{4} - 2 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q - 3 * q^2 + 5 * q^4 - 2 * q^7 - 12 * q^8 $$4 q - 3 q^{2} + 5 q^{4} - 2 q^{7} - 12 q^{8} + 2 q^{11} + 8 q^{13} + 3 q^{14} + 11 q^{16} - 10 q^{17} - 2 q^{19} - 3 q^{22} - 12 q^{23} + 7 q^{26} + 9 q^{28} + 4 q^{29} - 4 q^{31} - 17 q^{32} - q^{34} + 16 q^{37} - 10 q^{38} + 12 q^{41} - 2 q^{43} - 9 q^{44} - 8 q^{46} - 12 q^{47} + 6 q^{49} + 3 q^{52} - 10 q^{53} - 3 q^{58} - 2 q^{59} - 26 q^{61} + 20 q^{62} + 34 q^{64} - 2 q^{67} + 9 q^{68} + 10 q^{71} - 48 q^{74} + 16 q^{76} - 34 q^{77} + 22 q^{79} - 38 q^{82} - 10 q^{83} + 4 q^{86} + 4 q^{89} - 8 q^{91} + 28 q^{92} + 39 q^{94} + 22 q^{97} + 34 q^{98}+O(q^{100})$$ 4 * q - 3 * q^2 + 5 * q^4 - 2 * q^7 - 12 * q^8 + 2 * q^11 + 8 * q^13 + 3 * q^14 + 11 * q^16 - 10 * q^17 - 2 * q^19 - 3 * q^22 - 12 * q^23 + 7 * q^26 + 9 * q^28 + 4 * q^29 - 4 * q^31 - 17 * q^32 - q^34 + 16 * q^37 - 10 * q^38 + 12 * q^41 - 2 * q^43 - 9 * q^44 - 8 * q^46 - 12 * q^47 + 6 * q^49 + 3 * q^52 - 10 * q^53 - 3 * q^58 - 2 * q^59 - 26 * q^61 + 20 * q^62 + 34 * q^64 - 2 * q^67 + 9 * q^68 + 10 * q^71 - 48 * q^74 + 16 * q^76 - 34 * q^77 + 22 * q^79 - 38 * q^82 - 10 * q^83 + 4 * q^86 + 4 * q^89 - 8 * q^91 + 28 * q^92 + 39 * q^94 + 22 * q^97 + 34 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 1$$ v^3 - v^2 - 3*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 2$$ b3 + b2 + 4*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
 −1.75660 −0.820249 1.13856 2.43828
−2.75660 0 5.59883 0 0 −0.393832 −9.92054 0 0
1.2 −1.82025 0 1.31331 0 0 0.729126 1.24995 0 0
1.3 0.138564 0 −1.98080 0 0 −5.07830 −0.551597 0 0
1.4 1.43828 0 0.0686587 0 0 2.74301 −2.77782 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.bi 4
3.b odd 2 1 2175.2.a.v 4
5.b even 2 1 1305.2.a.r 4
15.d odd 2 1 435.2.a.j 4
15.e even 4 2 2175.2.c.n 8
60.h even 2 1 6960.2.a.co 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.j 4 15.d odd 2 1
1305.2.a.r 4 5.b even 2 1
2175.2.a.v 4 3.b odd 2 1
2175.2.c.n 8 15.e even 4 2
6525.2.a.bi 4 1.a even 1 1 trivial
6960.2.a.co 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6525))$$:

 $$T_{2}^{4} + 3T_{2}^{3} - 2T_{2}^{2} - 7T_{2} + 1$$ T2^4 + 3*T2^3 - 2*T2^2 - 7*T2 + 1 $$T_{7}^{4} + 2T_{7}^{3} - 15T_{7}^{2} + 4T_{7} + 4$$ T7^4 + 2*T7^3 - 15*T7^2 + 4*T7 + 4 $$T_{11}^{4} - 2T_{11}^{3} - 15T_{11}^{2} - 4T_{11} + 4$$ T11^4 - 2*T11^3 - 15*T11^2 - 4*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + \cdots + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2 T^{3} + \cdots + 4$$
$11$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$13$ $$T^{4} - 8 T^{3} + \cdots - 164$$
$17$ $$T^{4} + 10 T^{3} + \cdots - 116$$
$19$ $$T^{4} + 2 T^{3} + \cdots - 16$$
$23$ $$T^{4} + 12 T^{3} + \cdots - 1024$$
$29$ $$(T - 1)^{4}$$
$31$ $$T^{4} + 4 T^{3} + \cdots + 64$$
$37$ $$T^{4} - 16 T^{3} + \cdots - 2624$$
$41$ $$T^{4} - 12 T^{3} + \cdots - 1616$$
$43$ $$T^{4} + 2 T^{3} + \cdots + 1216$$
$47$ $$T^{4} + 12 T^{3} + \cdots - 64$$
$53$ $$T^{4} + 10 T^{3} + \cdots + 400$$
$59$ $$T^{4} + 2 T^{3} + \cdots + 10496$$
$61$ $$T^{4} + 26 T^{3} + \cdots - 11344$$
$67$ $$T^{4} + 2 T^{3} + \cdots - 124$$
$71$ $$T^{4} - 10 T^{3} + \cdots + 64$$
$73$ $$T^{4} - 84 T^{2} + \cdots - 256$$
$79$ $$T^{4} - 22 T^{3} + \cdots + 2416$$
$83$ $$T^{4} + 10 T^{3} + \cdots - 16$$
$89$ $$T^{4} - 4 T^{3} + \cdots + 10156$$
$97$ $$T^{4} - 22 T^{3} + \cdots + 2384$$