# Properties

 Label 1521.4.a.j 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + q^{4} - 9 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10})$$ q + 3 * q^2 + q^4 - 9 * q^5 - 2 * q^7 - 21 * q^8 $$q + 3 q^{2} + q^{4} - 9 q^{5} - 2 q^{7} - 21 q^{8} - 27 q^{10} + 30 q^{11} - 6 q^{14} - 71 q^{16} + 111 q^{17} + 46 q^{19} - 9 q^{20} + 90 q^{22} + 6 q^{23} - 44 q^{25} - 2 q^{28} + 105 q^{29} + 100 q^{31} - 45 q^{32} + 333 q^{34} + 18 q^{35} - 17 q^{37} + 138 q^{38} + 189 q^{40} - 231 q^{41} - 514 q^{43} + 30 q^{44} + 18 q^{46} - 162 q^{47} - 339 q^{49} - 132 q^{50} - 639 q^{53} - 270 q^{55} + 42 q^{56} + 315 q^{58} + 600 q^{59} + 233 q^{61} + 300 q^{62} + 433 q^{64} - 926 q^{67} + 111 q^{68} + 54 q^{70} - 930 q^{71} + 253 q^{73} - 51 q^{74} + 46 q^{76} - 60 q^{77} - 1324 q^{79} + 639 q^{80} - 693 q^{82} + 810 q^{83} - 999 q^{85} - 1542 q^{86} - 630 q^{88} + 498 q^{89} + 6 q^{92} - 486 q^{94} - 414 q^{95} - 1358 q^{97} - 1017 q^{98}+O(q^{100})$$ q + 3 * q^2 + q^4 - 9 * q^5 - 2 * q^7 - 21 * q^8 - 27 * q^10 + 30 * q^11 - 6 * q^14 - 71 * q^16 + 111 * q^17 + 46 * q^19 - 9 * q^20 + 90 * q^22 + 6 * q^23 - 44 * q^25 - 2 * q^28 + 105 * q^29 + 100 * q^31 - 45 * q^32 + 333 * q^34 + 18 * q^35 - 17 * q^37 + 138 * q^38 + 189 * q^40 - 231 * q^41 - 514 * q^43 + 30 * q^44 + 18 * q^46 - 162 * q^47 - 339 * q^49 - 132 * q^50 - 639 * q^53 - 270 * q^55 + 42 * q^56 + 315 * q^58 + 600 * q^59 + 233 * q^61 + 300 * q^62 + 433 * q^64 - 926 * q^67 + 111 * q^68 + 54 * q^70 - 930 * q^71 + 253 * q^73 - 51 * q^74 + 46 * q^76 - 60 * q^77 - 1324 * q^79 + 639 * q^80 - 693 * q^82 + 810 * q^83 - 999 * q^85 - 1542 * q^86 - 630 * q^88 + 498 * q^89 + 6 * q^92 - 486 * q^94 - 414 * q^95 - 1358 * q^97 - 1017 * 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
3.00000 0 1.00000 −9.00000 0 −2.00000 −21.0000 0 −27.0000
 $$n$$: 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.j 1
3.b odd 2 1 507.4.a.a 1
13.b even 2 1 1521.4.a.c 1
13.e even 6 2 117.4.g.b 2
39.d odd 2 1 507.4.a.e 1
39.f even 4 2 507.4.b.c 2
39.h odd 6 2 39.4.e.a 2
156.r even 6 2 624.4.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 39.h odd 6 2
117.4.g.b 2 13.e even 6 2
507.4.a.a 1 3.b odd 2 1
507.4.a.e 1 39.d odd 2 1
507.4.b.c 2 39.f even 4 2
624.4.q.b 2 156.r even 6 2
1521.4.a.c 1 13.b even 2 1
1521.4.a.j 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2} - 3$$ T2 - 3 $$T_{5} + 9$$ T5 + 9 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T$$
$5$ $$T + 9$$
$7$ $$T + 2$$
$11$ $$T - 30$$
$13$ $$T$$
$17$ $$T - 111$$
$19$ $$T - 46$$
$23$ $$T - 6$$
$29$ $$T - 105$$
$31$ $$T - 100$$
$37$ $$T + 17$$
$41$ $$T + 231$$
$43$ $$T + 514$$
$47$ $$T + 162$$
$53$ $$T + 639$$
$59$ $$T - 600$$
$61$ $$T - 233$$
$67$ $$T + 926$$
$71$ $$T + 930$$
$73$ $$T - 253$$
$79$ $$T + 1324$$
$83$ $$T - 810$$
$89$ $$T - 498$$
$97$ $$T + 1358$$