# Properties

 Label 1521.4.a.b 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 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 8 q^{4} - 17 q^{5} + 20 q^{7}+O(q^{10})$$ q - 4 * q^2 + 8 * q^4 - 17 * q^5 + 20 * q^7 $$q - 4 q^{2} + 8 q^{4} - 17 q^{5} + 20 q^{7} + 68 q^{10} + 32 q^{11} - 80 q^{14} - 64 q^{16} + 13 q^{17} + 30 q^{19} - 136 q^{20} - 128 q^{22} - 78 q^{23} + 164 q^{25} + 160 q^{28} - 197 q^{29} - 74 q^{31} + 256 q^{32} - 52 q^{34} - 340 q^{35} - 227 q^{37} - 120 q^{38} + 165 q^{41} - 156 q^{43} + 256 q^{44} + 312 q^{46} + 162 q^{47} + 57 q^{49} - 656 q^{50} - 93 q^{53} - 544 q^{55} + 788 q^{58} + 864 q^{59} + 145 q^{61} + 296 q^{62} - 512 q^{64} + 862 q^{67} + 104 q^{68} + 1360 q^{70} - 654 q^{71} + 215 q^{73} + 908 q^{74} + 240 q^{76} + 640 q^{77} - 76 q^{79} + 1088 q^{80} - 660 q^{82} - 628 q^{83} - 221 q^{85} + 624 q^{86} + 266 q^{89} - 624 q^{92} - 648 q^{94} - 510 q^{95} + 238 q^{97} - 228 q^{98}+O(q^{100})$$ q - 4 * q^2 + 8 * q^4 - 17 * q^5 + 20 * q^7 + 68 * q^10 + 32 * q^11 - 80 * q^14 - 64 * q^16 + 13 * q^17 + 30 * q^19 - 136 * q^20 - 128 * q^22 - 78 * q^23 + 164 * q^25 + 160 * q^28 - 197 * q^29 - 74 * q^31 + 256 * q^32 - 52 * q^34 - 340 * q^35 - 227 * q^37 - 120 * q^38 + 165 * q^41 - 156 * q^43 + 256 * q^44 + 312 * q^46 + 162 * q^47 + 57 * q^49 - 656 * q^50 - 93 * q^53 - 544 * q^55 + 788 * q^58 + 864 * q^59 + 145 * q^61 + 296 * q^62 - 512 * q^64 + 862 * q^67 + 104 * q^68 + 1360 * q^70 - 654 * q^71 + 215 * q^73 + 908 * q^74 + 240 * q^76 + 640 * q^77 - 76 * q^79 + 1088 * q^80 - 660 * q^82 - 628 * q^83 - 221 * q^85 + 624 * q^86 + 266 * q^89 - 624 * q^92 - 648 * q^94 - 510 * q^95 + 238 * q^97 - 228 * 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
−4.00000 0 8.00000 −17.0000 0 20.0000 0 0 68.0000
 $$n$$: 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.b 1
3.b odd 2 1 169.4.a.d 1
13.b even 2 1 1521.4.a.k 1
13.c even 3 2 117.4.g.c 2
39.d odd 2 1 169.4.a.a 1
39.f even 4 2 169.4.b.c 2
39.h odd 6 2 169.4.c.d 2
39.i odd 6 2 13.4.c.a 2
39.k even 12 4 169.4.e.c 4
156.p even 6 2 208.4.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 39.i odd 6 2
117.4.g.c 2 13.c even 3 2
169.4.a.a 1 39.d odd 2 1
169.4.a.d 1 3.b odd 2 1
169.4.b.c 2 39.f even 4 2
169.4.c.d 2 39.h odd 6 2
169.4.e.c 4 39.k even 12 4
208.4.i.b 2 156.p even 6 2
1521.4.a.b 1 1.a even 1 1 trivial
1521.4.a.k 1 13.b even 2 1

## Hecke kernels

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

 $$T_{2} + 4$$ T2 + 4 $$T_{5} + 17$$ T5 + 17 $$T_{7} - 20$$ T7 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T$$
$5$ $$T + 17$$
$7$ $$T - 20$$
$11$ $$T - 32$$
$13$ $$T$$
$17$ $$T - 13$$
$19$ $$T - 30$$
$23$ $$T + 78$$
$29$ $$T + 197$$
$31$ $$T + 74$$
$37$ $$T + 227$$
$41$ $$T - 165$$
$43$ $$T + 156$$
$47$ $$T - 162$$
$53$ $$T + 93$$
$59$ $$T - 864$$
$61$ $$T - 145$$
$67$ $$T - 862$$
$71$ $$T + 654$$
$73$ $$T - 215$$
$79$ $$T + 76$$
$83$ $$T + 628$$
$89$ $$T - 266$$
$97$ $$T - 238$$