# Properties

 Label 1521.4.a.y Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $4$ CM discriminant -39 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.8112.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 3$$ x^4 - 5*x^2 + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 117) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{2} + 8) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (-b2 + 8) * q^4 + (-b3 + 2*b1) * q^5 + (b3 - b1) * q^8 $$q - \beta_1 q^{2} + ( - \beta_{2} + 8) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8} + (17 \beta_{2} - 29) q^{10} + (3 \beta_{3} - 8 \beta_1) q^{11} + ( - 8 \beta_{2} - 51) q^{16} + ( - 9 \beta_{3} + 30 \beta_1) q^{20} + ( - 53 \beta_{2} + 119) q^{22} + ( - 46 \beta_{2} + 125) q^{25} + 51 \beta_1 q^{32} + (29 \beta_{2} - 221) q^{40} + ( - 23 \beta_{3} + 50 \beta_1) q^{41} - 452 q^{43} + (29 \beta_{3} - 108 \beta_1) q^{44} + (23 \beta_{3} + 76 \beta_1) q^{47} - 343 q^{49} + (46 \beta_{3} - 171 \beta_1) q^{50} + (172 \beta_{2} - 808) q^{55} + (\beta_{3} + 160 \beta_1) q^{59} + 262 \beta_{2} q^{61} + (115 \beta_{2} - 408) q^{64} + (43 \beta_{3} + 140 \beta_1) q^{71} - 116 \beta_{2} q^{79} + (43 \beta_{3} + 10 \beta_1) q^{80} + (395 \beta_{2} - 731) q^{82} + (51 \beta_{3} - 208 \beta_1) q^{83} + 452 \beta_1 q^{86} + ( - 119 \beta_{2} + 689) q^{88} + ( - 61 \beta_{3} - 194 \beta_1) q^{89} + ( - 269 \beta_{2} - 1285) q^{94} + 343 \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 + (-b2 + 8) * q^4 + (-b3 + 2*b1) * q^5 + (b3 - b1) * q^8 + (17*b2 - 29) * q^10 + (3*b3 - 8*b1) * q^11 + (-8*b2 - 51) * q^16 + (-9*b3 + 30*b1) * q^20 + (-53*b2 + 119) * q^22 + (-46*b2 + 125) * q^25 + 51*b1 * q^32 + (29*b2 - 221) * q^40 + (-23*b3 + 50*b1) * q^41 - 452 * q^43 + (29*b3 - 108*b1) * q^44 + (23*b3 + 76*b1) * q^47 - 343 * q^49 + (46*b3 - 171*b1) * q^50 + (172*b2 - 808) * q^55 + (b3 + 160*b1) * q^59 + 262*b2 * q^61 + (115*b2 - 408) * q^64 + (43*b3 + 140*b1) * q^71 - 116*b2 * q^79 + (43*b3 + 10*b1) * q^80 + (395*b2 - 731) * q^82 + (51*b3 - 208*b1) * q^83 + 452*b1 * q^86 + (-119*b2 + 689) * q^88 + (-61*b3 - 194*b1) * q^89 + (-269*b2 - 1285) * q^94 + 343*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{4}+O(q^{10})$$ 4 * q + 32 * q^4 $$4 q + 32 q^{4} - 116 q^{10} - 204 q^{16} + 476 q^{22} + 500 q^{25} - 884 q^{40} - 1808 q^{43} - 1372 q^{49} - 3232 q^{55} - 1632 q^{64} - 2924 q^{82} + 2756 q^{88} - 5140 q^{94}+O(q^{100})$$ 4 * q + 32 * q^4 - 116 * q^10 - 204 * q^16 + 476 * q^22 + 500 * q^25 - 884 * q^40 - 1808 * q^43 - 1372 * q^49 - 3232 * q^55 - 1632 * q^64 - 2924 * q^82 + 2756 * q^88 - 5140 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 3$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{3} + 6\nu$$ -v^3 + 6*v $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 5$$ 2*v^2 - 5 $$\beta_{3}$$ $$=$$ $$6\nu^{3} - 18\nu$$ 6*v^3 - 18*v
 $$\nu$$ $$=$$ $$( \beta_{3} + 6\beta_1 ) / 18$$ (b3 + 6*b1) / 18 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 5 ) / 2$$ (b2 + 5) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + 3\beta_1 ) / 3$$ (b3 + 3*b1) / 3

## 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
 0.835000 2.07431 −2.07431 −0.835000
−4.42782 0 11.6056 20.3925 0 0 −15.9647 0 −90.2944
1.2 −3.52058 0 4.39445 −9.17304 0 0 12.6936 0 32.2944
1.3 3.52058 0 4.39445 9.17304 0 0 −12.6936 0 32.2944
1.4 4.42782 0 11.6056 −20.3925 0 0 15.9647 0 −90.2944
 $$n$$: 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
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 1521.4.a.y 4
3.b odd 2 1 inner 1521.4.a.y 4
13.b even 2 1 inner 1521.4.a.y 4
13.d odd 4 2 117.4.b.c 4
39.d odd 2 1 CM 1521.4.a.y 4
39.f even 4 2 117.4.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.b.c 4 13.d odd 4 2
117.4.b.c 4 39.f even 4 2
1521.4.a.y 4 1.a even 1 1 trivial
1521.4.a.y 4 3.b odd 2 1 inner
1521.4.a.y 4 13.b even 2 1 inner
1521.4.a.y 4 39.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{4} - 32T_{2}^{2} + 243$$ T2^4 - 32*T2^2 + 243 $$T_{5}^{4} - 500T_{5}^{2} + 34992$$ T5^4 - 500*T5^2 + 34992 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 32T^{2} + 243$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 500 T^{2} + 34992$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 5324 T^{2} + \cdots + 2056752$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4} - 275684 T^{2} + \cdots + 9184890672$$
$43$ $$(T + 452)^{4}$$
$47$ $$T^{4} - 415292 T^{2} + \cdots + 2077595568$$
$53$ $$T^{4}$$
$59$ $$T^{4} - 821516 T^{2} + \cdots + 163107544752$$
$61$ $$(T^{2} - 892372)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4} - 1431644 T^{2} + \cdots + 21709693872$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 174928)^{2}$$
$83$ $$T^{4} - 2287148 T^{2} + \cdots + 20406871728$$
$89$ $$T^{4} - 2819876 T^{2} + \cdots + 67381252272$$
$97$ $$T^{4}$$