# Properties

 Label 1521.4.a.o Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 5 q^{4} + \beta q^{5} + 8 \beta q^{7} - 13 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 5 * q^4 + b * q^5 + 8*b * q^7 - 13*b * q^8 $$q + \beta q^{2} - 5 q^{4} + \beta q^{5} + 8 \beta q^{7} - 13 \beta q^{8} + 3 q^{10} - 8 \beta q^{11} + 24 q^{14} + q^{16} + 117 q^{17} - 66 \beta q^{19} - 5 \beta q^{20} - 24 q^{22} - 78 q^{23} - 122 q^{25} - 40 \beta q^{28} + 141 q^{29} + 90 \beta q^{31} + 105 \beta q^{32} + 117 \beta q^{34} + 24 q^{35} - 83 \beta q^{37} - 198 q^{38} - 39 q^{40} + 157 \beta q^{41} - 104 q^{43} + 40 \beta q^{44} - 78 \beta q^{46} + 174 \beta q^{47} - 151 q^{49} - 122 \beta q^{50} - 93 q^{53} - 24 q^{55} - 312 q^{56} + 141 \beta q^{58} - 164 \beta q^{59} + 145 q^{61} + 270 q^{62} + 307 q^{64} + 454 \beta q^{67} - 585 q^{68} + 24 \beta q^{70} - 610 \beta q^{71} - 265 \beta q^{73} - 249 q^{74} + 330 \beta q^{76} - 192 q^{77} + 1276 q^{79} + \beta q^{80} + 471 q^{82} + 456 \beta q^{83} + 117 \beta q^{85} - 104 \beta q^{86} + 312 q^{88} + 564 \beta q^{89} + 390 q^{92} + 522 q^{94} - 198 q^{95} + 116 \beta q^{97} - 151 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 5 * q^4 + b * q^5 + 8*b * q^7 - 13*b * q^8 + 3 * q^10 - 8*b * q^11 + 24 * q^14 + q^16 + 117 * q^17 - 66*b * q^19 - 5*b * q^20 - 24 * q^22 - 78 * q^23 - 122 * q^25 - 40*b * q^28 + 141 * q^29 + 90*b * q^31 + 105*b * q^32 + 117*b * q^34 + 24 * q^35 - 83*b * q^37 - 198 * q^38 - 39 * q^40 + 157*b * q^41 - 104 * q^43 + 40*b * q^44 - 78*b * q^46 + 174*b * q^47 - 151 * q^49 - 122*b * q^50 - 93 * q^53 - 24 * q^55 - 312 * q^56 + 141*b * q^58 - 164*b * q^59 + 145 * q^61 + 270 * q^62 + 307 * q^64 + 454*b * q^67 - 585 * q^68 + 24*b * q^70 - 610*b * q^71 - 265*b * q^73 - 249 * q^74 + 330*b * q^76 - 192 * q^77 + 1276 * q^79 + b * q^80 + 471 * q^82 + 456*b * q^83 + 117*b * q^85 - 104*b * q^86 + 312 * q^88 + 564*b * q^89 + 390 * q^92 + 522 * q^94 - 198 * q^95 + 116*b * q^97 - 151*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{4}+O(q^{10})$$ 2 * q - 10 * q^4 $$2 q - 10 q^{4} + 6 q^{10} + 48 q^{14} + 2 q^{16} + 234 q^{17} - 48 q^{22} - 156 q^{23} - 244 q^{25} + 282 q^{29} + 48 q^{35} - 396 q^{38} - 78 q^{40} - 208 q^{43} - 302 q^{49} - 186 q^{53} - 48 q^{55} - 624 q^{56} + 290 q^{61} + 540 q^{62} + 614 q^{64} - 1170 q^{68} - 498 q^{74} - 384 q^{77} + 2552 q^{79} + 942 q^{82} + 624 q^{88} + 780 q^{92} + 1044 q^{94} - 396 q^{95}+O(q^{100})$$ 2 * q - 10 * q^4 + 6 * q^10 + 48 * q^14 + 2 * q^16 + 234 * q^17 - 48 * q^22 - 156 * q^23 - 244 * q^25 + 282 * q^29 + 48 * q^35 - 396 * q^38 - 78 * q^40 - 208 * q^43 - 302 * q^49 - 186 * q^53 - 48 * q^55 - 624 * q^56 + 290 * q^61 + 540 * q^62 + 614 * q^64 - 1170 * q^68 - 498 * q^74 - 384 * q^77 + 2552 * q^79 + 942 * q^82 + 624 * q^88 + 780 * q^92 + 1044 * q^94 - 396 * q^95

## 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
 −1.73205 1.73205
−1.73205 0 −5.00000 −1.73205 0 −13.8564 22.5167 0 3.00000
1.2 1.73205 0 −5.00000 1.73205 0 13.8564 −22.5167 0 3.00000
 $$n$$: 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.o 2
3.b odd 2 1 169.4.a.i 2
13.b even 2 1 inner 1521.4.a.o 2
13.f odd 12 2 117.4.q.a 2
39.d odd 2 1 169.4.a.i 2
39.f even 4 2 169.4.b.d 2
39.h odd 6 2 169.4.c.h 4
39.i odd 6 2 169.4.c.h 4
39.k even 12 2 13.4.e.b 2
39.k even 12 2 169.4.e.a 2
156.v odd 12 2 208.4.w.b 2

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

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{7}^{2} - 192$$ T7^2 - 192

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3$$
$7$ $$T^{2} - 192$$
$11$ $$T^{2} - 192$$
$13$ $$T^{2}$$
$17$ $$(T - 117)^{2}$$
$19$ $$T^{2} - 13068$$
$23$ $$(T + 78)^{2}$$
$29$ $$(T - 141)^{2}$$
$31$ $$T^{2} - 24300$$
$37$ $$T^{2} - 20667$$
$41$ $$T^{2} - 73947$$
$43$ $$(T + 104)^{2}$$
$47$ $$T^{2} - 90828$$
$53$ $$(T + 93)^{2}$$
$59$ $$T^{2} - 80688$$
$61$ $$(T - 145)^{2}$$
$67$ $$T^{2} - 618348$$
$71$ $$T^{2} - 1116300$$
$73$ $$T^{2} - 210675$$
$79$ $$(T - 1276)^{2}$$
$83$ $$T^{2} - 623808$$
$89$ $$T^{2} - 954288$$
$97$ $$T^{2} - 40368$$