# Properties

 Label 6525.2.a.p 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + (2 \beta + 2) q^{7} + (\beta - 3) q^{8}+O(q^{10})$$ q + (b - 1) * q^2 + (-2*b + 1) * q^4 + (2*b + 2) * q^7 + (b - 3) * q^8 $$q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + (2 \beta + 2) q^{7} + (\beta - 3) q^{8} + ( - 2 \beta + 2) q^{11} + 2 q^{13} + 2 q^{14} + 3 q^{16} + 2 \beta q^{17} + ( - 2 \beta - 2) q^{19} + (4 \beta - 6) q^{22} + (2 \beta - 6) q^{23} + (2 \beta - 2) q^{26} + ( - 2 \beta - 6) q^{28} - q^{29} + (6 \beta - 2) q^{31} + (\beta + 3) q^{32} + ( - 2 \beta + 4) q^{34} + 6 \beta q^{37} - 2 q^{38} + 6 q^{41} + 6 q^{43} + ( - 6 \beta + 10) q^{44} + ( - 8 \beta + 10) q^{46} + ( - 4 \beta - 6) q^{47} + (8 \beta + 5) q^{49} + ( - 4 \beta + 2) q^{52} + ( - 4 \beta + 2) q^{53} + ( - 4 \beta - 2) q^{56} + ( - \beta + 1) q^{58} + ( - 4 \beta + 2) q^{61} + ( - 8 \beta + 14) q^{62} + (2 \beta - 7) q^{64} + ( - 6 \beta + 2) q^{67} + (2 \beta - 8) q^{68} + (8 \beta + 4) q^{71} - 6 \beta q^{73} + ( - 6 \beta + 12) q^{74} + (2 \beta + 6) q^{76} - 4 q^{77} + ( - 6 \beta + 6) q^{79} + (6 \beta - 6) q^{82} + ( - 2 \beta + 10) q^{83} + (6 \beta - 6) q^{86} + (8 \beta - 10) q^{88} + (4 \beta + 2) q^{89} + (4 \beta + 4) q^{91} + (14 \beta - 14) q^{92} + ( - 2 \beta - 2) q^{94} + (6 \beta + 4) q^{97} + ( - 3 \beta + 11) q^{98} +O(q^{100})$$ q + (b - 1) * q^2 + (-2*b + 1) * q^4 + (2*b + 2) * q^7 + (b - 3) * q^8 + (-2*b + 2) * q^11 + 2 * q^13 + 2 * q^14 + 3 * q^16 + 2*b * q^17 + (-2*b - 2) * q^19 + (4*b - 6) * q^22 + (2*b - 6) * q^23 + (2*b - 2) * q^26 + (-2*b - 6) * q^28 - q^29 + (6*b - 2) * q^31 + (b + 3) * q^32 + (-2*b + 4) * q^34 + 6*b * q^37 - 2 * q^38 + 6 * q^41 + 6 * q^43 + (-6*b + 10) * q^44 + (-8*b + 10) * q^46 + (-4*b - 6) * q^47 + (8*b + 5) * q^49 + (-4*b + 2) * q^52 + (-4*b + 2) * q^53 + (-4*b - 2) * q^56 + (-b + 1) * q^58 + (-4*b + 2) * q^61 + (-8*b + 14) * q^62 + (2*b - 7) * q^64 + (-6*b + 2) * q^67 + (2*b - 8) * q^68 + (8*b + 4) * q^71 - 6*b * q^73 + (-6*b + 12) * q^74 + (2*b + 6) * q^76 - 4 * q^77 + (-6*b + 6) * q^79 + (6*b - 6) * q^82 + (-2*b + 10) * q^83 + (6*b - 6) * q^86 + (8*b - 10) * q^88 + (4*b + 2) * q^89 + (4*b + 4) * q^91 + (14*b - 14) * q^92 + (-2*b - 2) * q^94 + (6*b + 4) * q^97 + (-3*b + 11) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} + 4 q^{11} + 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 12 q^{22} - 12 q^{23} - 4 q^{26} - 12 q^{28} - 2 q^{29} - 4 q^{31} + 6 q^{32} + 8 q^{34} - 4 q^{38} + 12 q^{41} + 12 q^{43} + 20 q^{44} + 20 q^{46} - 12 q^{47} + 10 q^{49} + 4 q^{52} + 4 q^{53} - 4 q^{56} + 2 q^{58} + 4 q^{61} + 28 q^{62} - 14 q^{64} + 4 q^{67} - 16 q^{68} + 8 q^{71} + 24 q^{74} + 12 q^{76} - 8 q^{77} + 12 q^{79} - 12 q^{82} + 20 q^{83} - 12 q^{86} - 20 q^{88} + 4 q^{89} + 8 q^{91} - 28 q^{92} - 4 q^{94} + 8 q^{97} + 22 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 + 4 * q^11 + 4 * q^13 + 4 * q^14 + 6 * q^16 - 4 * q^19 - 12 * q^22 - 12 * q^23 - 4 * q^26 - 12 * q^28 - 2 * q^29 - 4 * q^31 + 6 * q^32 + 8 * q^34 - 4 * q^38 + 12 * q^41 + 12 * q^43 + 20 * q^44 + 20 * q^46 - 12 * q^47 + 10 * q^49 + 4 * q^52 + 4 * q^53 - 4 * q^56 + 2 * q^58 + 4 * q^61 + 28 * q^62 - 14 * q^64 + 4 * q^67 - 16 * q^68 + 8 * q^71 + 24 * q^74 + 12 * q^76 - 8 * q^77 + 12 * q^79 - 12 * q^82 + 20 * q^83 - 12 * q^86 - 20 * q^88 + 4 * q^89 + 8 * q^91 - 28 * q^92 - 4 * q^94 + 8 * q^97 + 22 * 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
 −1.41421 1.41421
−2.41421 0 3.82843 0 0 −0.828427 −4.41421 0 0
1.2 0.414214 0 −1.82843 0 0 4.82843 −1.58579 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.p 2
3.b odd 2 1 725.2.a.c 2
5.b even 2 1 1305.2.a.n 2
15.d odd 2 1 145.2.a.b 2
15.e even 4 2 725.2.b.c 4
60.h even 2 1 2320.2.a.k 2
105.g even 2 1 7105.2.a.e 2
120.i odd 2 1 9280.2.a.be 2
120.m even 2 1 9280.2.a.w 2
435.b odd 2 1 4205.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.b 2 15.d odd 2 1
725.2.a.c 2 3.b odd 2 1
725.2.b.c 4 15.e even 4 2
1305.2.a.n 2 5.b even 2 1
2320.2.a.k 2 60.h even 2 1
4205.2.a.d 2 435.b odd 2 1
6525.2.a.p 2 1.a even 1 1 trivial
7105.2.a.e 2 105.g even 2 1
9280.2.a.w 2 120.m even 2 1
9280.2.a.be 2 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6525))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T - 4$$
$11$ $$T^{2} - 4T - 4$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 8$$
$19$ $$T^{2} + 4T - 4$$
$23$ $$T^{2} + 12T + 28$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} + 4T - 68$$
$37$ $$T^{2} - 72$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} + 12T + 4$$
$53$ $$T^{2} - 4T - 28$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 4T - 28$$
$67$ $$T^{2} - 4T - 68$$
$71$ $$T^{2} - 8T - 112$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} - 12T - 36$$
$83$ $$T^{2} - 20T + 92$$
$89$ $$T^{2} - 4T - 28$$
$97$ $$T^{2} - 8T - 56$$