# Properties

 Label 6525.2.a.j Level $6525$ Weight $2$ Character orbit 6525.a Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

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 $$q + q^{2} - q^{4} - 4 q^{7} - 3 q^{8} + 4 q^{11} - 6 q^{13} - 4 q^{14} - q^{16} + 6 q^{17} - 4 q^{19} + 4 q^{22} - 4 q^{23} - 6 q^{26} + 4 q^{28} - q^{29} - 8 q^{31} + 5 q^{32} + 6 q^{34} - 2 q^{37} - 4 q^{38} + 6 q^{41} - 4 q^{43} - 4 q^{44} - 4 q^{46} + 9 q^{49} + 6 q^{52} - 10 q^{53} + 12 q^{56} - q^{58} + 12 q^{59} - 10 q^{61} - 8 q^{62} + 7 q^{64} - 8 q^{67} - 6 q^{68} + 8 q^{71} + 2 q^{73} - 2 q^{74} + 4 q^{76} - 16 q^{77} + 6 q^{82} + 8 q^{83} - 4 q^{86} - 12 q^{88} + 6 q^{89} + 24 q^{91} + 4 q^{92} + 2 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 - q^4 - 4 * q^7 - 3 * q^8 + 4 * q^11 - 6 * q^13 - 4 * q^14 - q^16 + 6 * q^17 - 4 * q^19 + 4 * q^22 - 4 * q^23 - 6 * q^26 + 4 * q^28 - q^29 - 8 * q^31 + 5 * q^32 + 6 * q^34 - 2 * q^37 - 4 * q^38 + 6 * q^41 - 4 * q^43 - 4 * q^44 - 4 * q^46 + 9 * q^49 + 6 * q^52 - 10 * q^53 + 12 * q^56 - q^58 + 12 * q^59 - 10 * q^61 - 8 * q^62 + 7 * q^64 - 8 * q^67 - 6 * q^68 + 8 * q^71 + 2 * q^73 - 2 * q^74 + 4 * q^76 - 16 * q^77 + 6 * q^82 + 8 * q^83 - 4 * q^86 - 12 * q^88 + 6 * q^89 + 24 * q^91 + 4 * q^92 + 2 * 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.j 1
3.b odd 2 1 2175.2.a.b 1
5.b even 2 1 1305.2.a.b 1
15.d odd 2 1 435.2.a.d 1
15.e even 4 2 2175.2.c.b 2
60.h even 2 1 6960.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.d 1 15.d odd 2 1
1305.2.a.b 1 5.b even 2 1
2175.2.a.b 1 3.b odd 2 1
2175.2.c.b 2 15.e even 4 2
6525.2.a.j 1 1.a even 1 1 trivial
6960.2.a.l 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 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T - 4$$
$13$ $$T + 6$$
$17$ $$T - 6$$
$19$ $$T + 4$$
$23$ $$T + 4$$
$29$ $$T + 1$$
$31$ $$T + 8$$
$37$ $$T + 2$$
$41$ $$T - 6$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T + 10$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T + 8$$
$71$ $$T - 8$$
$73$ $$T - 2$$
$79$ $$T$$
$83$ $$T - 8$$
$89$ $$T - 6$$
$97$ $$T - 2$$
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