# Properties

 Label 1521.4.a.d Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + q^{4} + 9 q^{5} + 15 q^{7} + 21 q^{8}+O(q^{10})$$ q - 3 * q^2 + q^4 + 9 * q^5 + 15 * q^7 + 21 * q^8 $$q - 3 q^{2} + q^{4} + 9 q^{5} + 15 q^{7} + 21 q^{8} - 27 q^{10} + 48 q^{11} - 45 q^{14} - 71 q^{16} - 45 q^{17} + 6 q^{19} + 9 q^{20} - 144 q^{22} + 162 q^{23} - 44 q^{25} + 15 q^{28} + 144 q^{29} + 264 q^{31} + 45 q^{32} + 135 q^{34} + 135 q^{35} + 303 q^{37} - 18 q^{38} + 189 q^{40} + 192 q^{41} + 97 q^{43} + 48 q^{44} - 486 q^{46} - 111 q^{47} - 118 q^{49} + 132 q^{50} + 414 q^{53} + 432 q^{55} + 315 q^{56} - 432 q^{58} - 522 q^{59} + 376 q^{61} - 792 q^{62} + 433 q^{64} - 36 q^{67} - 45 q^{68} - 405 q^{70} - 357 q^{71} - 1098 q^{73} - 909 q^{74} + 6 q^{76} + 720 q^{77} - 830 q^{79} - 639 q^{80} - 576 q^{82} + 438 q^{83} - 405 q^{85} - 291 q^{86} + 1008 q^{88} + 438 q^{89} + 162 q^{92} + 333 q^{94} + 54 q^{95} - 852 q^{97} + 354 q^{98}+O(q^{100})$$ q - 3 * q^2 + q^4 + 9 * q^5 + 15 * q^7 + 21 * q^8 - 27 * q^10 + 48 * q^11 - 45 * q^14 - 71 * q^16 - 45 * q^17 + 6 * q^19 + 9 * q^20 - 144 * q^22 + 162 * q^23 - 44 * q^25 + 15 * q^28 + 144 * q^29 + 264 * q^31 + 45 * q^32 + 135 * q^34 + 135 * q^35 + 303 * q^37 - 18 * q^38 + 189 * q^40 + 192 * q^41 + 97 * q^43 + 48 * q^44 - 486 * q^46 - 111 * q^47 - 118 * q^49 + 132 * q^50 + 414 * q^53 + 432 * q^55 + 315 * q^56 - 432 * q^58 - 522 * q^59 + 376 * q^61 - 792 * q^62 + 433 * q^64 - 36 * q^67 - 45 * q^68 - 405 * q^70 - 357 * q^71 - 1098 * q^73 - 909 * q^74 + 6 * q^76 + 720 * q^77 - 830 * q^79 - 639 * q^80 - 576 * q^82 + 438 * q^83 - 405 * q^85 - 291 * q^86 + 1008 * q^88 + 438 * q^89 + 162 * q^92 + 333 * q^94 + 54 * q^95 - 852 * q^97 + 354 * 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 15.0000 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.d 1
3.b odd 2 1 169.4.a.c 1
13.b even 2 1 1521.4.a.i 1
13.d odd 4 2 117.4.b.a 2
39.d odd 2 1 169.4.a.b 1
39.f even 4 2 13.4.b.a 2
39.h odd 6 2 169.4.c.c 2
39.i odd 6 2 169.4.c.b 2
39.k even 12 4 169.4.e.d 4
156.l odd 4 2 208.4.f.b 2
195.j odd 4 2 325.4.d.b 2
195.n even 4 2 325.4.c.b 2
195.u odd 4 2 325.4.d.a 2
312.w odd 4 2 832.4.f.c 2
312.y even 4 2 832.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 39.f even 4 2
117.4.b.a 2 13.d odd 4 2
169.4.a.b 1 39.d odd 2 1
169.4.a.c 1 3.b odd 2 1
169.4.c.b 2 39.i odd 6 2
169.4.c.c 2 39.h odd 6 2
169.4.e.d 4 39.k even 12 4
208.4.f.b 2 156.l odd 4 2
325.4.c.b 2 195.n even 4 2
325.4.d.a 2 195.u odd 4 2
325.4.d.b 2 195.j odd 4 2
832.4.f.c 2 312.w odd 4 2
832.4.f.e 2 312.y even 4 2
1521.4.a.d 1 1.a even 1 1 trivial
1521.4.a.i 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{5} - 9$$ T5 - 9 $$T_{7} - 15$$ T7 - 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T - 9$$
$7$ $$T - 15$$
$11$ $$T - 48$$
$13$ $$T$$
$17$ $$T + 45$$
$19$ $$T - 6$$
$23$ $$T - 162$$
$29$ $$T - 144$$
$31$ $$T - 264$$
$37$ $$T - 303$$
$41$ $$T - 192$$
$43$ $$T - 97$$
$47$ $$T + 111$$
$53$ $$T - 414$$
$59$ $$T + 522$$
$61$ $$T - 376$$
$67$ $$T + 36$$
$71$ $$T + 357$$
$73$ $$T + 1098$$
$79$ $$T + 830$$
$83$ $$T - 438$$
$89$ $$T - 438$$
$97$ $$T + 852$$