# Properties

 Label 1521.2.a.k Level $1521$ Weight $2$ Character orbit 1521.a Self dual yes Analytic conductor $12.145$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$12.1452461474$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} - \beta q^{5} - \beta q^{8} +O(q^{10})$$ q + b * q^2 + q^4 - b * q^5 - b * q^8 $$q + \beta q^{2} + q^{4} - \beta q^{5} - \beta q^{8} - 3 q^{10} - 5 q^{16} - 3 q^{17} + 2 \beta q^{19} - \beta q^{20} - 6 q^{23} - 2 q^{25} - 3 q^{29} - 2 \beta q^{31} - 3 \beta q^{32} - 3 \beta q^{34} - 5 \beta q^{37} + 6 q^{38} + 3 q^{40} + 3 \beta q^{41} - 8 q^{43} - 6 \beta q^{46} + 2 \beta q^{47} - 7 q^{49} - 2 \beta q^{50} + 3 q^{53} - 3 \beta q^{58} - 4 \beta q^{59} + q^{61} - 6 q^{62} + q^{64} + 2 \beta q^{67} - 3 q^{68} + 2 \beta q^{71} + \beta q^{73} - 15 q^{74} + 2 \beta q^{76} + 4 q^{79} + 5 \beta q^{80} + 9 q^{82} + 8 \beta q^{83} + 3 \beta q^{85} - 8 \beta q^{86} - 4 \beta q^{89} - 6 q^{92} + 6 q^{94} - 6 q^{95} - 4 \beta q^{97} - 7 \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^4 - b * q^5 - b * q^8 - 3 * q^10 - 5 * q^16 - 3 * q^17 + 2*b * q^19 - b * q^20 - 6 * q^23 - 2 * q^25 - 3 * q^29 - 2*b * q^31 - 3*b * q^32 - 3*b * q^34 - 5*b * q^37 + 6 * q^38 + 3 * q^40 + 3*b * q^41 - 8 * q^43 - 6*b * q^46 + 2*b * q^47 - 7 * q^49 - 2*b * q^50 + 3 * q^53 - 3*b * q^58 - 4*b * q^59 + q^61 - 6 * q^62 + q^64 + 2*b * q^67 - 3 * q^68 + 2*b * q^71 + b * q^73 - 15 * q^74 + 2*b * q^76 + 4 * q^79 + 5*b * q^80 + 9 * q^82 + 8*b * q^83 + 3*b * q^85 - 8*b * q^86 - 4*b * q^89 - 6 * q^92 + 6 * q^94 - 6 * q^95 - 4*b * q^97 - 7*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} - 6 q^{10} - 10 q^{16} - 6 q^{17} - 12 q^{23} - 4 q^{25} - 6 q^{29} + 12 q^{38} + 6 q^{40} - 16 q^{43} - 14 q^{49} + 6 q^{53} + 2 q^{61} - 12 q^{62} + 2 q^{64} - 6 q^{68} - 30 q^{74} + 8 q^{79} + 18 q^{82} - 12 q^{92} + 12 q^{94} - 12 q^{95}+O(q^{100})$$ 2 * q + 2 * q^4 - 6 * q^10 - 10 * q^16 - 6 * q^17 - 12 * q^23 - 4 * q^25 - 6 * q^29 + 12 * q^38 + 6 * q^40 - 16 * q^43 - 14 * q^49 + 6 * q^53 + 2 * q^61 - 12 * q^62 + 2 * q^64 - 6 * q^68 - 30 * q^74 + 8 * q^79 + 18 * q^82 - 12 * q^92 + 12 * q^94 - 12 * q^95

## 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
 −1.73205 1.73205
−1.73205 0 1.00000 1.73205 0 0 1.73205 0 −3.00000
1.2 1.73205 0 1.00000 −1.73205 0 0 −1.73205 0 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.a.k 2
3.b odd 2 1 169.2.a.a 2
12.b even 2 1 2704.2.a.o 2
13.b even 2 1 inner 1521.2.a.k 2
13.d odd 4 2 1521.2.b.a 2
13.f odd 12 2 117.2.q.c 2
15.d odd 2 1 4225.2.a.v 2
21.c even 2 1 8281.2.a.q 2
39.d odd 2 1 169.2.a.a 2
39.f even 4 2 169.2.b.a 2
39.h odd 6 2 169.2.c.a 4
39.i odd 6 2 169.2.c.a 4
39.k even 12 2 13.2.e.a 2
39.k even 12 2 169.2.e.a 2
52.l even 12 2 1872.2.by.d 2
156.h even 2 1 2704.2.a.o 2
156.l odd 4 2 2704.2.f.b 2
156.v odd 12 2 208.2.w.b 2
195.e odd 2 1 4225.2.a.v 2
195.bc odd 12 2 325.2.m.a 4
195.bh even 12 2 325.2.n.a 2
195.bn odd 12 2 325.2.m.a 4
273.g even 2 1 8281.2.a.q 2
273.bs odd 12 2 637.2.k.c 2
273.bv even 12 2 637.2.k.a 2
273.bw even 12 2 637.2.u.c 2
273.ca odd 12 2 637.2.q.a 2
273.ch odd 12 2 637.2.u.b 2
312.bo even 12 2 832.2.w.d 2
312.bq odd 12 2 832.2.w.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 39.k even 12 2
117.2.q.c 2 13.f odd 12 2
169.2.a.a 2 3.b odd 2 1
169.2.a.a 2 39.d odd 2 1
169.2.b.a 2 39.f even 4 2
169.2.c.a 4 39.h odd 6 2
169.2.c.a 4 39.i odd 6 2
169.2.e.a 2 39.k even 12 2
208.2.w.b 2 156.v odd 12 2
325.2.m.a 4 195.bc odd 12 2
325.2.m.a 4 195.bn odd 12 2
325.2.n.a 2 195.bh even 12 2
637.2.k.a 2 273.bv even 12 2
637.2.k.c 2 273.bs odd 12 2
637.2.q.a 2 273.ca odd 12 2
637.2.u.b 2 273.ch odd 12 2
637.2.u.c 2 273.bw even 12 2
832.2.w.a 2 312.bq odd 12 2
832.2.w.d 2 312.bo even 12 2
1521.2.a.k 2 1.a even 1 1 trivial
1521.2.a.k 2 13.b even 2 1 inner
1521.2.b.a 2 13.d odd 4 2
1872.2.by.d 2 52.l even 12 2
2704.2.a.o 2 12.b even 2 1
2704.2.a.o 2 156.h even 2 1
2704.2.f.b 2 156.l odd 4 2
4225.2.a.v 2 15.d odd 2 1
4225.2.a.v 2 195.e odd 2 1
8281.2.a.q 2 21.c even 2 1
8281.2.a.q 2 273.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} - 12$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T + 3)^{2}$$
$31$ $$T^{2} - 12$$
$37$ $$T^{2} - 75$$
$41$ $$T^{2} - 27$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} - 12$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2} - 48$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} - 12$$
$71$ $$T^{2} - 12$$
$73$ $$T^{2} - 3$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} - 192$$
$89$ $$T^{2} - 48$$
$97$ $$T^{2} - 48$$