# Properties

 Label 6525.2.a.t Level $6525$ Weight $2$ Character orbit 6525.a Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta + 3) q^{4} - q^{7} + ( - 2 \beta - 5) q^{8} +O(q^{10})$$ q - b * q^2 + (b + 3) * q^4 - q^7 + (-2*b - 5) * q^8 $$q - \beta q^{2} + (\beta + 3) q^{4} - q^{7} + ( - 2 \beta - 5) q^{8} - 5 q^{11} + ( - 2 \beta + 1) q^{13} + \beta q^{14} + (5 \beta + 4) q^{16} - 3 q^{17} - 2 \beta q^{19} + 5 \beta q^{22} - 4 q^{23} + (\beta + 10) q^{26} + ( - \beta - 3) q^{28} - q^{29} + 4 q^{31} + ( - 5 \beta - 15) q^{32} + 3 \beta q^{34} + 4 q^{37} + (2 \beta + 10) q^{38} + ( - 4 \beta + 2) q^{41} + ( - 2 \beta + 6) q^{43} + ( - 5 \beta - 15) q^{44} + 4 \beta q^{46} + ( - 2 \beta + 7) q^{47} - 6 q^{49} + ( - 7 \beta - 7) q^{52} + (2 \beta + 4) q^{53} + (2 \beta + 5) q^{56} + \beta q^{58} + ( - 2 \beta + 4) q^{59} + ( - 6 \beta + 2) q^{61} - 4 \beta q^{62} + (10 \beta + 17) q^{64} + ( - 4 \beta - 3) q^{67} + ( - 3 \beta - 9) q^{68} + ( - 2 \beta + 6) q^{71} - 4 q^{73} - 4 \beta q^{74} + ( - 8 \beta - 10) q^{76} + 5 q^{77} + ( - 2 \beta + 4) q^{79} + (2 \beta + 20) q^{82} + (2 \beta - 8) q^{83} + ( - 4 \beta + 10) q^{86} + (10 \beta + 25) q^{88} + ( - 2 \beta - 5) q^{89} + (2 \beta - 1) q^{91} + ( - 4 \beta - 12) q^{92} + ( - 5 \beta + 10) q^{94} + (2 \beta - 8) q^{97} + 6 \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b + 3) * q^4 - q^7 + (-2*b - 5) * q^8 - 5 * q^11 + (-2*b + 1) * q^13 + b * q^14 + (5*b + 4) * q^16 - 3 * q^17 - 2*b * q^19 + 5*b * q^22 - 4 * q^23 + (b + 10) * q^26 + (-b - 3) * q^28 - q^29 + 4 * q^31 + (-5*b - 15) * q^32 + 3*b * q^34 + 4 * q^37 + (2*b + 10) * q^38 + (-4*b + 2) * q^41 + (-2*b + 6) * q^43 + (-5*b - 15) * q^44 + 4*b * q^46 + (-2*b + 7) * q^47 - 6 * q^49 + (-7*b - 7) * q^52 + (2*b + 4) * q^53 + (2*b + 5) * q^56 + b * q^58 + (-2*b + 4) * q^59 + (-6*b + 2) * q^61 - 4*b * q^62 + (10*b + 17) * q^64 + (-4*b - 3) * q^67 + (-3*b - 9) * q^68 + (-2*b + 6) * q^71 - 4 * q^73 - 4*b * q^74 + (-8*b - 10) * q^76 + 5 * q^77 + (-2*b + 4) * q^79 + (2*b + 20) * q^82 + (2*b - 8) * q^83 + (-4*b + 10) * q^86 + (10*b + 25) * q^88 + (-2*b - 5) * q^89 + (2*b - 1) * q^91 + (-4*b - 12) * q^92 + (-5*b + 10) * q^94 + (2*b - 8) * q^97 + 6*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 7 q^{4} - 2 q^{7} - 12 q^{8}+O(q^{10})$$ 2 * q - q^2 + 7 * q^4 - 2 * q^7 - 12 * q^8 $$2 q - q^{2} + 7 q^{4} - 2 q^{7} - 12 q^{8} - 10 q^{11} + q^{14} + 13 q^{16} - 6 q^{17} - 2 q^{19} + 5 q^{22} - 8 q^{23} + 21 q^{26} - 7 q^{28} - 2 q^{29} + 8 q^{31} - 35 q^{32} + 3 q^{34} + 8 q^{37} + 22 q^{38} + 10 q^{43} - 35 q^{44} + 4 q^{46} + 12 q^{47} - 12 q^{49} - 21 q^{52} + 10 q^{53} + 12 q^{56} + q^{58} + 6 q^{59} - 2 q^{61} - 4 q^{62} + 44 q^{64} - 10 q^{67} - 21 q^{68} + 10 q^{71} - 8 q^{73} - 4 q^{74} - 28 q^{76} + 10 q^{77} + 6 q^{79} + 42 q^{82} - 14 q^{83} + 16 q^{86} + 60 q^{88} - 12 q^{89} - 28 q^{92} + 15 q^{94} - 14 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q - q^2 + 7 * q^4 - 2 * q^7 - 12 * q^8 - 10 * q^11 + q^14 + 13 * q^16 - 6 * q^17 - 2 * q^19 + 5 * q^22 - 8 * q^23 + 21 * q^26 - 7 * q^28 - 2 * q^29 + 8 * q^31 - 35 * q^32 + 3 * q^34 + 8 * q^37 + 22 * q^38 + 10 * q^43 - 35 * q^44 + 4 * q^46 + 12 * q^47 - 12 * q^49 - 21 * q^52 + 10 * q^53 + 12 * q^56 + q^58 + 6 * q^59 - 2 * q^61 - 4 * q^62 + 44 * q^64 - 10 * q^67 - 21 * q^68 + 10 * q^71 - 8 * q^73 - 4 * q^74 - 28 * q^76 + 10 * q^77 + 6 * q^79 + 42 * q^82 - 14 * q^83 + 16 * q^86 + 60 * q^88 - 12 * q^89 - 28 * q^92 + 15 * q^94 - 14 * q^97 + 6 * q^98

## 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
 2.79129 −1.79129
−2.79129 0 5.79129 0 0 −1.00000 −10.5826 0 0
1.2 1.79129 0 1.20871 0 0 −1.00000 −1.41742 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.t 2
3.b odd 2 1 2175.2.a.r 2
5.b even 2 1 1305.2.a.m 2
15.d odd 2 1 435.2.a.f 2
15.e even 4 2 2175.2.c.f 4
60.h even 2 1 6960.2.a.bw 2

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

 $$T_{2}^{2} + T_{2} - 5$$ T2^2 + T2 - 5 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 5$$ T11 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 5$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T + 5)^{2}$$
$13$ $$T^{2} - 21$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 2T - 20$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T + 1)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} - 84$$
$43$ $$T^{2} - 10T + 4$$
$47$ $$T^{2} - 12T + 15$$
$53$ $$T^{2} - 10T + 4$$
$59$ $$T^{2} - 6T - 12$$
$61$ $$T^{2} + 2T - 188$$
$67$ $$T^{2} + 10T - 59$$
$71$ $$T^{2} - 10T + 4$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} - 6T - 12$$
$83$ $$T^{2} + 14T + 28$$
$89$ $$T^{2} + 12T + 15$$
$97$ $$T^{2} + 14T + 28$$