# Properties

 Label 1521.4.a.x Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: 4.4.5054412.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 29x^{2} + 48$$ x^4 - 29*x^2 + 48 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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)

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + 7) q^{4} - \beta_{2} q^{5} + 6 \beta_1 q^{7} + (2 \beta_{2} + 9 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + 7) * q^4 - b2 * q^5 + 6*b1 * q^7 + (2*b2 + 9*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + 7) q^{4} - \beta_{2} q^{5} + 6 \beta_1 q^{7} + (2 \beta_{2} + 9 \beta_1) q^{8} + ( - 2 \beta_{3} - 6) q^{10} + ( - \beta_{2} - 2 \beta_1) q^{11} + (6 \beta_{3} + 90) q^{14} + (5 \beta_{3} + 91) q^{16} + 54 q^{17} + ( - 6 \beta_{2} - 6 \beta_1) q^{19} + (4 \beta_{2} - 26 \beta_1) q^{20} + ( - 4 \beta_{3} - 36) q^{22} + (12 \beta_{3} + 36) q^{23} + ( - 8 \beta_{3} + 7) q^{25} + (12 \beta_{2} + 102 \beta_1) q^{28} + (12 \beta_{3} + 18) q^{29} + ( - 6 \beta_{2} + 18 \beta_1) q^{31} + ( - 6 \beta_{2} + 69 \beta_1) q^{32} + 54 \beta_1 q^{34} + ( - 12 \beta_{3} - 36) q^{35} + ( - 24 \beta_{2} + 48 \beta_1) q^{37} + ( - 18 \beta_{3} - 126) q^{38} + ( - 2 \beta_{3} - 318) q^{40} + (13 \beta_{2} - 4 \beta_1) q^{41} + (12 \beta_{3} - 260) q^{43} - 60 \beta_1 q^{44} + (24 \beta_{2} + 156 \beta_1) q^{46} + ( - 21 \beta_{2} + 122 \beta_1) q^{47} + (36 \beta_{3} + 197) q^{49} + ( - 16 \beta_{2} - 73 \beta_1) q^{50} + ( - 36 \beta_{3} + 198) q^{53} + ( - 4 \beta_{3} + 144) q^{55} + (78 \beta_{3} + 882) q^{56} + (24 \beta_{2} + 138 \beta_1) q^{58} + ( - 15 \beta_{2} - 34 \beta_1) q^{59} + (8 \beta_{3} - 566) q^{61} + (6 \beta_{3} + 234) q^{62} + (17 \beta_{3} + 271) q^{64} + ( - 12 \beta_{2} - 150 \beta_1) q^{67} + (54 \beta_{3} + 378) q^{68} + ( - 24 \beta_{2} - 156 \beta_1) q^{70} + (23 \beta_{2} + 6 \beta_1) q^{71} - 54 \beta_{2} q^{73} + 576 q^{74} + (12 \beta_{2} - 258 \beta_1) q^{76} + ( - 24 \beta_{3} - 216) q^{77} + (32 \beta_{3} + 88) q^{79} + ( - 36 \beta_{2} - 130 \beta_1) q^{80} + (22 \beta_{3} + 18) q^{82} + ( - 25 \beta_{2} + 138 \beta_1) q^{83} - 54 \beta_{2} q^{85} + (24 \beta_{2} - 140 \beta_1) q^{86} + ( - 28 \beta_{3} - 612) q^{88} + (15 \beta_{2} - 4 \beta_1) q^{89} + (108 \beta_{3} + 2196) q^{92} + (80 \beta_{3} + 1704) q^{94} + ( - 36 \beta_{3} + 828) q^{95} + ( - 54 \beta_{2} + 336 \beta_1) q^{97} + (72 \beta_{2} + 557 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 + 7) * q^4 - b2 * q^5 + 6*b1 * q^7 + (2*b2 + 9*b1) * q^8 + (-2*b3 - 6) * q^10 + (-b2 - 2*b1) * q^11 + (6*b3 + 90) * q^14 + (5*b3 + 91) * q^16 + 54 * q^17 + (-6*b2 - 6*b1) * q^19 + (4*b2 - 26*b1) * q^20 + (-4*b3 - 36) * q^22 + (12*b3 + 36) * q^23 + (-8*b3 + 7) * q^25 + (12*b2 + 102*b1) * q^28 + (12*b3 + 18) * q^29 + (-6*b2 + 18*b1) * q^31 + (-6*b2 + 69*b1) * q^32 + 54*b1 * q^34 + (-12*b3 - 36) * q^35 + (-24*b2 + 48*b1) * q^37 + (-18*b3 - 126) * q^38 + (-2*b3 - 318) * q^40 + (13*b2 - 4*b1) * q^41 + (12*b3 - 260) * q^43 - 60*b1 * q^44 + (24*b2 + 156*b1) * q^46 + (-21*b2 + 122*b1) * q^47 + (36*b3 + 197) * q^49 + (-16*b2 - 73*b1) * q^50 + (-36*b3 + 198) * q^53 + (-4*b3 + 144) * q^55 + (78*b3 + 882) * q^56 + (24*b2 + 138*b1) * q^58 + (-15*b2 - 34*b1) * q^59 + (8*b3 - 566) * q^61 + (6*b3 + 234) * q^62 + (17*b3 + 271) * q^64 + (-12*b2 - 150*b1) * q^67 + (54*b3 + 378) * q^68 + (-24*b2 - 156*b1) * q^70 + (23*b2 + 6*b1) * q^71 - 54*b2 * q^73 + 576 * q^74 + (12*b2 - 258*b1) * q^76 + (-24*b3 - 216) * q^77 + (32*b3 + 88) * q^79 + (-36*b2 - 130*b1) * q^80 + (22*b3 + 18) * q^82 + (-25*b2 + 138*b1) * q^83 - 54*b2 * q^85 + (24*b2 - 140*b1) * q^86 + (-28*b3 - 612) * q^88 + (15*b2 - 4*b1) * q^89 + (108*b3 + 2196) * q^92 + (80*b3 + 1704) * q^94 + (-36*b3 + 828) * q^95 + (-54*b2 + 336*b1) * q^97 + (72*b2 + 557*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 26 q^{4}+O(q^{10})$$ 4 * q + 26 * q^4 $$4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} + 1050 q^{64} + 1404 q^{68} + 2304 q^{74} - 816 q^{77} + 288 q^{79} + 28 q^{82} - 2392 q^{88} + 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100})$$ 4 * q + 26 * q^4 - 20 * q^10 + 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 + 44 * q^25 + 48 * q^29 - 120 * q^35 - 468 * q^38 - 1268 * q^40 - 1064 * q^43 + 716 * q^49 + 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 + 1050 * q^64 + 1404 * q^68 + 2304 * q^74 - 816 * q^77 + 288 * q^79 + 28 * q^82 - 2392 * q^88 + 8568 * q^92 + 6656 * q^94 + 3384 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 29x^{2} + 48$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 25\nu ) / 2$$ (v^3 - 25*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 15$$ v^2 - 15
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 15$$ b3 + 15 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 25\beta_1$$ 2*b2 + 25*b1

## 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
 −5.21898 −1.32750 1.32750 5.21898
−5.21898 0 19.2377 5.83936 0 −31.3139 −58.6495 0 −30.4755
1.2 −1.32750 0 −6.23774 −15.4241 0 −7.96501 18.9006 0 20.4755
1.3 1.32750 0 −6.23774 15.4241 0 7.96501 −18.9006 0 20.4755
1.4 5.21898 0 19.2377 −5.83936 0 31.3139 58.6495 0 −30.4755
 $$n$$: 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
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.x 4
3.b odd 2 1 507.4.a.j 4
13.b even 2 1 inner 1521.4.a.x 4
13.d odd 4 2 117.4.b.d 4
39.d odd 2 1 507.4.a.j 4
39.f even 4 2 39.4.b.a 4
156.l odd 4 2 624.4.c.e 4

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

## Hecke kernels

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

 $$T_{2}^{4} - 29T_{2}^{2} + 48$$ T2^4 - 29*T2^2 + 48 $$T_{5}^{4} - 272T_{5}^{2} + 8112$$ T5^4 - 272*T5^2 + 8112 $$T_{7}^{4} - 1044T_{7}^{2} + 62208$$ T7^4 - 1044*T7^2 + 62208

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 29T^{2} + 48$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 272T^{2} + 8112$$
$7$ $$T^{4} - 1044 T^{2} + 62208$$
$11$ $$T^{4} - 428 T^{2} + 43200$$
$13$ $$T^{4}$$
$17$ $$(T - 54)^{4}$$
$19$ $$T^{4} - 11556 T^{2} + \cdots + 31492800$$
$23$ $$(T^{2} - 60 T - 22464)^{2}$$
$29$ $$(T^{2} - 24 T - 23220)^{2}$$
$31$ $$T^{4} - 17028 T^{2} + \cdots + 47044800$$
$37$ $$T^{4} - 200448 T^{2} + \cdots + 2293235712$$
$41$ $$T^{4} - 45392 T^{2} + \cdots + 128314800$$
$43$ $$(T^{2} + 532 T + 47392)^{2}$$
$47$ $$T^{4} - 500348 T^{2} + \cdots + 62387841792$$
$53$ $$(T^{2} - 432 T - 163620)^{2}$$
$59$ $$T^{4} - 104924 T^{2} + \cdots + 2436066048$$
$61$ $$(T^{2} + 1140 T + 314516)^{2}$$
$67$ $$T^{4} - 727668 T^{2} + \cdots + 143327232$$
$71$ $$T^{4} - 147692 T^{2} + \cdots + 3298756800$$
$73$ $$T^{4} - 793152 T^{2} + \cdots + 68976790272$$
$79$ $$(T^{2} - 144 T - 160960)^{2}$$
$83$ $$T^{4} - 653276 T^{2} + \cdots + 106682723328$$
$89$ $$T^{4} - 60464 T^{2} + \cdots + 249304368$$
$97$ $$T^{4} - 3704256 T^{2} + \cdots + 3383532000000$$