# Properties

 Label 1521.4.a.q 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{2} + 4 q^{4} + 8 \beta q^{5} + 13 \beta q^{7} - 8 \beta q^{8} +O(q^{10})$$ q + 2*b * q^2 + 4 * q^4 + 8*b * q^5 + 13*b * q^7 - 8*b * q^8 $$q + 2 \beta q^{2} + 4 q^{4} + 8 \beta q^{5} + 13 \beta q^{7} - 8 \beta q^{8} + 48 q^{10} - 13 \beta q^{11} + 78 q^{14} - 80 q^{16} + 27 q^{17} + 51 \beta q^{19} + 32 \beta q^{20} - 78 q^{22} + 57 q^{23} + 67 q^{25} + 52 \beta q^{28} + 69 q^{29} + 42 \beta q^{31} - 96 \beta q^{32} + 54 \beta q^{34} + 312 q^{35} + 23 \beta q^{37} + 306 q^{38} - 192 q^{40} + 227 \beta q^{41} + 85 q^{43} - 52 \beta q^{44} + 114 \beta q^{46} + 198 \beta q^{47} + 164 q^{49} + 134 \beta q^{50} - 426 q^{53} - 312 q^{55} - 312 q^{56} + 138 \beta q^{58} + 11 \beta q^{59} - 17 q^{61} + 252 q^{62} + 64 q^{64} + 95 \beta q^{67} + 108 q^{68} + 624 \beta q^{70} + 337 \beta q^{71} + 580 \beta q^{73} + 138 q^{74} + 204 \beta q^{76} - 507 q^{77} - 1244 q^{79} - 640 \beta q^{80} + 1362 q^{82} - 246 \beta q^{83} + 216 \beta q^{85} + 170 \beta q^{86} + 312 q^{88} + 177 \beta q^{89} + 228 q^{92} + 1188 q^{94} + 1224 q^{95} - 713 \beta q^{97} + 328 \beta q^{98} +O(q^{100})$$ q + 2*b * q^2 + 4 * q^4 + 8*b * q^5 + 13*b * q^7 - 8*b * q^8 + 48 * q^10 - 13*b * q^11 + 78 * q^14 - 80 * q^16 + 27 * q^17 + 51*b * q^19 + 32*b * q^20 - 78 * q^22 + 57 * q^23 + 67 * q^25 + 52*b * q^28 + 69 * q^29 + 42*b * q^31 - 96*b * q^32 + 54*b * q^34 + 312 * q^35 + 23*b * q^37 + 306 * q^38 - 192 * q^40 + 227*b * q^41 + 85 * q^43 - 52*b * q^44 + 114*b * q^46 + 198*b * q^47 + 164 * q^49 + 134*b * q^50 - 426 * q^53 - 312 * q^55 - 312 * q^56 + 138*b * q^58 + 11*b * q^59 - 17 * q^61 + 252 * q^62 + 64 * q^64 + 95*b * q^67 + 108 * q^68 + 624*b * q^70 + 337*b * q^71 + 580*b * q^73 + 138 * q^74 + 204*b * q^76 - 507 * q^77 - 1244 * q^79 - 640*b * q^80 + 1362 * q^82 - 246*b * q^83 + 216*b * q^85 + 170*b * q^86 + 312 * q^88 + 177*b * q^89 + 228 * q^92 + 1188 * q^94 + 1224 * q^95 - 713*b * q^97 + 328*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{4}+O(q^{10})$$ 2 * q + 8 * q^4 $$2 q + 8 q^{4} + 96 q^{10} + 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} + 134 q^{25} + 138 q^{29} + 624 q^{35} + 612 q^{38} - 384 q^{40} + 170 q^{43} + 328 q^{49} - 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} + 128 q^{64} + 216 q^{68} + 276 q^{74} - 1014 q^{77} - 2488 q^{79} + 2724 q^{82} + 624 q^{88} + 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100})$$ 2 * q + 8 * q^4 + 96 * q^10 + 156 * q^14 - 160 * q^16 + 54 * q^17 - 156 * q^22 + 114 * q^23 + 134 * q^25 + 138 * q^29 + 624 * q^35 + 612 * q^38 - 384 * q^40 + 170 * q^43 + 328 * q^49 - 852 * q^53 - 624 * q^55 - 624 * q^56 - 34 * q^61 + 504 * q^62 + 128 * q^64 + 216 * q^68 + 276 * q^74 - 1014 * q^77 - 2488 * q^79 + 2724 * q^82 + 624 * q^88 + 456 * q^92 + 2376 * q^94 + 2448 * 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
−3.46410 0 4.00000 −13.8564 0 −22.5167 13.8564 0 48.0000
1.2 3.46410 0 4.00000 13.8564 0 22.5167 −13.8564 0 48.0000
 $$n$$: 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

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.q 2
3.b odd 2 1 169.4.a.h 2
13.b even 2 1 inner 1521.4.a.q 2
13.f odd 12 2 117.4.q.c 2
39.d odd 2 1 169.4.a.h 2
39.f even 4 2 169.4.b.b 2
39.h odd 6 2 169.4.c.i 4
39.i odd 6 2 169.4.c.i 4
39.k even 12 2 13.4.e.a 2
39.k even 12 2 169.4.e.b 2
156.v odd 12 2 208.4.w.a 2

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

 $$T_{2}^{2} - 12$$ T2^2 - 12 $$T_{5}^{2} - 192$$ T5^2 - 192 $$T_{7}^{2} - 507$$ T7^2 - 507

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 12$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 192$$
$7$ $$T^{2} - 507$$
$11$ $$T^{2} - 507$$
$13$ $$T^{2}$$
$17$ $$(T - 27)^{2}$$
$19$ $$T^{2} - 7803$$
$23$ $$(T - 57)^{2}$$
$29$ $$(T - 69)^{2}$$
$31$ $$T^{2} - 5292$$
$37$ $$T^{2} - 1587$$
$41$ $$T^{2} - 154587$$
$43$ $$(T - 85)^{2}$$
$47$ $$T^{2} - 117612$$
$53$ $$(T + 426)^{2}$$
$59$ $$T^{2} - 363$$
$61$ $$(T + 17)^{2}$$
$67$ $$T^{2} - 27075$$
$71$ $$T^{2} - 340707$$
$73$ $$T^{2} - 1009200$$
$79$ $$(T + 1244)^{2}$$
$83$ $$T^{2} - 181548$$
$89$ $$T^{2} - 93987$$
$97$ $$T^{2} - 1525107$$