# Properties

 Label 6525.2.a.e Level $6525$ Weight $2$ Character orbit 6525.a Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 4 q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + 4 * q^7 + 3 * q^8 $$q - q^{2} - q^{4} + 4 q^{7} + 3 q^{8} - 6 q^{13} - 4 q^{14} - q^{16} + 2 q^{17} + 8 q^{19} - 4 q^{23} + 6 q^{26} - 4 q^{28} - q^{29} + 4 q^{31} - 5 q^{32} - 2 q^{34} - 6 q^{37} - 8 q^{38} - 2 q^{41} + 4 q^{43} + 4 q^{46} + 9 q^{49} + 6 q^{52} + 6 q^{53} + 12 q^{56} + q^{58} + 12 q^{59} + 6 q^{61} - 4 q^{62} + 7 q^{64} + 8 q^{67} - 2 q^{68} - 16 q^{71} + 6 q^{73} + 6 q^{74} - 8 q^{76} + 12 q^{79} + 2 q^{82} - 16 q^{83} - 4 q^{86} - 2 q^{89} - 24 q^{91} + 4 q^{92} + 14 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 - q^4 + 4 * q^7 + 3 * q^8 - 6 * q^13 - 4 * q^14 - q^16 + 2 * q^17 + 8 * q^19 - 4 * q^23 + 6 * q^26 - 4 * q^28 - q^29 + 4 * q^31 - 5 * q^32 - 2 * q^34 - 6 * q^37 - 8 * q^38 - 2 * q^41 + 4 * q^43 + 4 * q^46 + 9 * q^49 + 6 * q^52 + 6 * q^53 + 12 * q^56 + q^58 + 12 * q^59 + 6 * q^61 - 4 * q^62 + 7 * q^64 + 8 * q^67 - 2 * q^68 - 16 * q^71 + 6 * q^73 + 6 * q^74 - 8 * q^76 + 12 * q^79 + 2 * q^82 - 16 * q^83 - 4 * q^86 - 2 * q^89 - 24 * q^91 + 4 * q^92 + 14 * q^97 - 9 * 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
 0
−1.00000 0 −1.00000 0 0 4.00000 3.00000 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.e 1
3.b odd 2 1 2175.2.a.h 1
5.b even 2 1 1305.2.a.e 1
15.d odd 2 1 435.2.a.a 1
15.e even 4 2 2175.2.c.a 2
60.h even 2 1 6960.2.a.w 1

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

 $$T_{2} + 1$$ T2 + 1 $$T_{7} - 4$$ T7 - 4 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T$$
$13$ $$T + 6$$
$17$ $$T - 2$$
$19$ $$T - 8$$
$23$ $$T + 4$$
$29$ $$T + 1$$
$31$ $$T - 4$$
$37$ $$T + 6$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T - 6$$
$67$ $$T - 8$$
$71$ $$T + 16$$
$73$ $$T - 6$$
$79$ $$T - 12$$
$83$ $$T + 16$$
$89$ $$T + 2$$
$97$ $$T - 14$$