# Properties

 Label 1521.4.a.w 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: 4.4.1362828.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 23x^{2} + 48$$ x^4 - 23*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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + 4) q^{4} + (\beta_{2} + 2 \beta_1) q^{5} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{2} + 3 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + 4) * q^4 + (b2 + 2*b1) * q^5 + (-2*b2 - 2*b1) * q^7 + (2*b2 + 3*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + 4) q^{4} + (\beta_{2} + 2 \beta_1) q^{5} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{2} + 3 \beta_1) q^{8} + (4 \beta_{3} + 24) q^{10} + ( - 3 \beta_{2} + 8 \beta_1) q^{11} + ( - 6 \beta_{3} - 24) q^{14} + ( - \beta_{3} + 4) q^{16} + (12 \beta_{3} + 30) q^{17} + ( - 8 \beta_{2} + 10 \beta_1) q^{19} + 36 \beta_1 q^{20} + (2 \beta_{3} + 96) q^{22} - 72 q^{23} + (4 \beta_{3} + 7) q^{25} + (4 \beta_{2} - 50 \beta_1) q^{28} + (12 \beta_{3} + 102) q^{29} + (20 \beta_{2} + 38 \beta_1) q^{31} + ( - 18 \beta_{2} - 27 \beta_1) q^{32} + (24 \beta_{2} + 114 \beta_1) q^{34} - 216 q^{35} + (4 \beta_{2} - 68 \beta_1) q^{37} + ( - 6 \beta_{3} + 120) q^{38} + (4 \beta_{3} + 240) q^{40} + ( - 13 \beta_{2} + 66 \beta_1) q^{41} + (12 \beta_{3} + 328) q^{43} + (28 \beta_{2} + 46 \beta_1) q^{44} - 72 \beta_1 q^{46} + (\beta_{2} - 36 \beta_1) q^{47} + ( - 12 \beta_{3} + 41) q^{49} + (8 \beta_{2} + 35 \beta_1) q^{50} + (48 \beta_{3} - 222) q^{53} + (44 \beta_{3} - 60) q^{55} + (6 \beta_{3} - 408) q^{56} + (24 \beta_{2} + 186 \beta_1) q^{58} + ( - \beta_{2} + 52 \beta_1) q^{59} + ( - 28 \beta_{3} + 58) q^{61} + (78 \beta_{3} + 456) q^{62} + ( - 55 \beta_{3} - 356) q^{64} + ( - 54 \beta_{2} + 54 \beta_1) q^{67} + (66 \beta_{3} + 1128) q^{68} - 216 \beta_1 q^{70} + (41 \beta_{2} + 168 \beta_1) q^{71} + ( - 42 \beta_{2} + 156 \beta_1) q^{73} + ( - 60 \beta_{3} - 816) q^{74} + (52 \beta_{2} - 2 \beta_1) q^{76} + ( - 84 \beta_{3} + 312) q^{77} + ( - 16 \beta_{3} + 1072) q^{79} + (8 \beta_{2} - 20 \beta_1) q^{80} + (40 \beta_{3} + 792) q^{82} + ( - 63 \beta_{2} - 188 \beta_1) q^{83} + ( - 18 \beta_{2} + 396 \beta_1) q^{85} + (24 \beta_{2} + 412 \beta_1) q^{86} + (86 \beta_{3} - 216) q^{88} + (25 \beta_{2} - 138 \beta_1) q^{89} + ( - 72 \beta_{3} - 288) q^{92} + ( - 34 \beta_{3} - 432) q^{94} + (72 \beta_{3} - 432) q^{95} + (14 \beta_{2} + 68 \beta_1) q^{97} + ( - 24 \beta_{2} - 43 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 + 4) * q^4 + (b2 + 2*b1) * q^5 + (-2*b2 - 2*b1) * q^7 + (2*b2 + 3*b1) * q^8 + (4*b3 + 24) * q^10 + (-3*b2 + 8*b1) * q^11 + (-6*b3 - 24) * q^14 + (-b3 + 4) * q^16 + (12*b3 + 30) * q^17 + (-8*b2 + 10*b1) * q^19 + 36*b1 * q^20 + (2*b3 + 96) * q^22 - 72 * q^23 + (4*b3 + 7) * q^25 + (4*b2 - 50*b1) * q^28 + (12*b3 + 102) * q^29 + (20*b2 + 38*b1) * q^31 + (-18*b2 - 27*b1) * q^32 + (24*b2 + 114*b1) * q^34 - 216 * q^35 + (4*b2 - 68*b1) * q^37 + (-6*b3 + 120) * q^38 + (4*b3 + 240) * q^40 + (-13*b2 + 66*b1) * q^41 + (12*b3 + 328) * q^43 + (28*b2 + 46*b1) * q^44 - 72*b1 * q^46 + (b2 - 36*b1) * q^47 + (-12*b3 + 41) * q^49 + (8*b2 + 35*b1) * q^50 + (48*b3 - 222) * q^53 + (44*b3 - 60) * q^55 + (6*b3 - 408) * q^56 + (24*b2 + 186*b1) * q^58 + (-b2 + 52*b1) * q^59 + (-28*b3 + 58) * q^61 + (78*b3 + 456) * q^62 + (-55*b3 - 356) * q^64 + (-54*b2 + 54*b1) * q^67 + (66*b3 + 1128) * q^68 - 216*b1 * q^70 + (41*b2 + 168*b1) * q^71 + (-42*b2 + 156*b1) * q^73 + (-60*b3 - 816) * q^74 + (52*b2 - 2*b1) * q^76 + (-84*b3 + 312) * q^77 + (-16*b3 + 1072) * q^79 + (8*b2 - 20*b1) * q^80 + (40*b3 + 792) * q^82 + (-63*b2 - 188*b1) * q^83 + (-18*b2 + 396*b1) * q^85 + (24*b2 + 412*b1) * q^86 + (86*b3 - 216) * q^88 + (25*b2 - 138*b1) * q^89 + (-72*b3 - 288) * q^92 + (-34*b3 - 432) * q^94 + (72*b3 - 432) * q^95 + (14*b2 + 68*b1) * q^97 + (-24*b2 - 43*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 14 q^{4}+O(q^{10})$$ 4 * q + 14 * q^4 $$4 q + 14 q^{4} + 88 q^{10} - 84 q^{14} + 18 q^{16} + 96 q^{17} + 380 q^{22} - 288 q^{23} + 20 q^{25} + 384 q^{29} - 864 q^{35} + 492 q^{38} + 952 q^{40} + 1288 q^{43} + 188 q^{49} - 984 q^{53} - 328 q^{55} - 1644 q^{56} + 288 q^{61} + 1668 q^{62} - 1314 q^{64} + 4380 q^{68} - 3144 q^{74} + 1416 q^{77} + 4320 q^{79} + 3088 q^{82} - 1036 q^{88} - 1008 q^{92} - 1660 q^{94} - 1872 q^{95}+O(q^{100})$$ 4 * q + 14 * q^4 + 88 * q^10 - 84 * q^14 + 18 * q^16 + 96 * q^17 + 380 * q^22 - 288 * q^23 + 20 * q^25 + 384 * q^29 - 864 * q^35 + 492 * q^38 + 952 * q^40 + 1288 * q^43 + 188 * q^49 - 984 * q^53 - 328 * q^55 - 1644 * q^56 + 288 * q^61 + 1668 * q^62 - 1314 * q^64 + 4380 * q^68 - 3144 * q^74 + 1416 * q^77 + 4320 * q^79 + 3088 * q^82 - 1036 * q^88 - 1008 * q^92 - 1660 * q^94 - 1872 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 23x^{2} + 48$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 19\nu ) / 2$$ (v^3 - 19*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 12$$ v^2 - 12
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 12$$ b3 + 12 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 19\beta_1$$ 2*b2 + 19*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
 −4.54739 −1.52356 1.52356 4.54739
−4.54739 0 12.6788 −12.9118 0 16.7289 −21.2762 0 58.7151
1.2 −1.52356 0 −5.67878 9.65841 0 −22.3639 20.8404 0 −14.7151
1.3 1.52356 0 −5.67878 −9.65841 0 22.3639 −20.8404 0 −14.7151
1.4 4.54739 0 12.6788 12.9118 0 −16.7289 21.2762 0 58.7151
 $$n$$: 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.w 4
3.b odd 2 1 507.4.a.l 4
13.b even 2 1 inner 1521.4.a.w 4
13.d odd 4 2 117.4.b.e 4
39.d odd 2 1 507.4.a.l 4
39.f even 4 2 39.4.b.b 4
156.l odd 4 2 624.4.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.b 4 39.f even 4 2
117.4.b.e 4 13.d odd 4 2
507.4.a.l 4 3.b odd 2 1
507.4.a.l 4 39.d odd 2 1
624.4.c.c 4 156.l odd 4 2
1521.4.a.w 4 1.a even 1 1 trivial
1521.4.a.w 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{4} - 23T_{2}^{2} + 48$$ T2^4 - 23*T2^2 + 48 $$T_{5}^{4} - 260T_{5}^{2} + 15552$$ T5^4 - 260*T5^2 + 15552 $$T_{7}^{4} - 780T_{7}^{2} + 139968$$ T7^4 - 780*T7^2 + 139968

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 23T^{2} + 48$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 260 T^{2} + 15552$$
$7$ $$T^{4} - 780 T^{2} + 139968$$
$11$ $$T^{4} - 3152 T^{2} + \cdots + 1572528$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 48 T - 11556)^{2}$$
$19$ $$T^{4} - 13884 T^{2} + \cdots + 3048192$$
$23$ $$(T + 72)^{4}$$
$29$ $$(T^{2} - 192 T - 2916)^{2}$$
$31$ $$T^{4} - 100572 T^{2} + \cdots + 2389782528$$
$37$ $$T^{4} - 110256 T^{2} + \cdots + 2060577792$$
$41$ $$T^{4} - 133364 T^{2} + \cdots + 4431055872$$
$43$ $$(T^{2} - 644 T + 91552)^{2}$$
$47$ $$T^{4} - 30128 T^{2} + \cdots + 116663088$$
$53$ $$(T^{2} + 492 T - 133596)^{2}$$
$59$ $$T^{4} - 62576 T^{2} + \cdots + 457419312$$
$61$ $$(T^{2} - 144 T - 60868)^{2}$$
$67$ $$T^{4} - 591948 T^{2} + \cdots + 918330048$$
$71$ $$T^{4} - 917456 T^{2} + \cdots + 59484058032$$
$73$ $$T^{4} - 896400 T^{2} + \cdots + 179358354432$$
$79$ $$(T^{2} - 2160 T + 1144832)^{2}$$
$83$ $$T^{4} - 1464080 T^{2} + \cdots + 317023217328$$
$89$ $$T^{4} - 561812 T^{2} + \cdots + 78903164928$$
$97$ $$T^{4} - 137040 T^{2} + \cdots + 725594112$$