# Properties

 Label 1521.4.a.f Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $1$ 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] = [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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{4} - 12 q^{5} - 2 q^{7}+O(q^{10})$$ q - 8 * q^4 - 12 * q^5 - 2 * q^7 $$q - 8 q^{4} - 12 q^{5} - 2 q^{7} - 36 q^{11} + 64 q^{16} + 78 q^{17} - 74 q^{19} + 96 q^{20} + 96 q^{23} + 19 q^{25} + 16 q^{28} - 18 q^{29} + 214 q^{31} + 24 q^{35} + 286 q^{37} - 384 q^{41} + 524 q^{43} + 288 q^{44} + 300 q^{47} - 339 q^{49} - 558 q^{53} + 432 q^{55} + 576 q^{59} + 74 q^{61} - 512 q^{64} - 38 q^{67} - 624 q^{68} - 456 q^{71} + 682 q^{73} + 592 q^{76} + 72 q^{77} + 704 q^{79} - 768 q^{80} - 888 q^{83} - 936 q^{85} - 1020 q^{89} - 768 q^{92} + 888 q^{95} - 110 q^{97}+O(q^{100})$$ q - 8 * q^4 - 12 * q^5 - 2 * q^7 - 36 * q^11 + 64 * q^16 + 78 * q^17 - 74 * q^19 + 96 * q^20 + 96 * q^23 + 19 * q^25 + 16 * q^28 - 18 * q^29 + 214 * q^31 + 24 * q^35 + 286 * q^37 - 384 * q^41 + 524 * q^43 + 288 * q^44 + 300 * q^47 - 339 * q^49 - 558 * q^53 + 432 * q^55 + 576 * q^59 + 74 * q^61 - 512 * q^64 - 38 * q^67 - 624 * q^68 - 456 * q^71 + 682 * q^73 + 592 * q^76 + 72 * q^77 + 704 * q^79 - 768 * q^80 - 888 * q^83 - 936 * q^85 - 1020 * q^89 - 768 * q^92 + 888 * q^95 - 110 * q^97

## 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
0 0 −8.00000 −12.0000 0 −2.00000 0 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
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 1521.4.a.f 1
3.b odd 2 1 507.4.a.c 1
13.b even 2 1 117.4.a.a 1
39.d odd 2 1 39.4.a.a 1
39.f even 4 2 507.4.b.b 2
52.b odd 2 1 1872.4.a.m 1
156.h even 2 1 624.4.a.g 1
195.e odd 2 1 975.4.a.e 1
273.g even 2 1 1911.4.a.f 1
312.b odd 2 1 2496.4.a.o 1
312.h even 2 1 2496.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.a 1 39.d odd 2 1
117.4.a.a 1 13.b even 2 1
507.4.a.c 1 3.b odd 2 1
507.4.b.b 2 39.f even 4 2
624.4.a.g 1 156.h even 2 1
975.4.a.e 1 195.e odd 2 1
1521.4.a.f 1 1.a even 1 1 trivial
1872.4.a.m 1 52.b odd 2 1
1911.4.a.f 1 273.g even 2 1
2496.4.a.f 1 312.h even 2 1
2496.4.a.o 1 312.b odd 2 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5} + 12$$ T5 + 12 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 12$$
$7$ $$T + 2$$
$11$ $$T + 36$$
$13$ $$T$$
$17$ $$T - 78$$
$19$ $$T + 74$$
$23$ $$T - 96$$
$29$ $$T + 18$$
$31$ $$T - 214$$
$37$ $$T - 286$$
$41$ $$T + 384$$
$43$ $$T - 524$$
$47$ $$T - 300$$
$53$ $$T + 558$$
$59$ $$T - 576$$
$61$ $$T - 74$$
$67$ $$T + 38$$
$71$ $$T + 456$$
$73$ $$T - 682$$
$79$ $$T - 704$$
$83$ $$T + 888$$
$89$ $$T + 1020$$
$97$ $$T + 110$$