Properties

 Label 1521.2.a.n Level $1521$ Weight $2$ Character orbit 1521.a Self dual yes Analytic conductor $12.145$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,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, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$12.1452461474$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 507) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_1 + 4) q^{4} + (\beta_{2} - 2 \beta_1 - 1) q^{5} + (\beta_{2} + 1) q^{7} + (3 \beta_1 - 5) q^{8}+O(q^{10})$$ q + (b2 + b1 - 1) * q^2 + (-b1 + 4) * q^4 + (b2 - 2*b1 - 1) * q^5 + (b2 + 1) * q^7 + (3*b1 - 5) * q^8 $$q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_1 + 4) q^{4} + (\beta_{2} - 2 \beta_1 - 1) q^{5} + (\beta_{2} + 1) q^{7} + (3 \beta_1 - 5) q^{8} + ( - 6 \beta_{2} + 2 \beta_1 - 3) q^{10} + ( - 3 \beta_{2} + \beta_1 - 3) q^{11} + (2 \beta_1 + 1) q^{14} + (\beta_{2} - 6 \beta_1 + 6) q^{16} + (2 \beta_{2} - 3 \beta_1 + 2) q^{17} + ( - \beta_1 - 2) q^{19} + (5 \beta_{2} - 7 \beta_1 - 1) q^{20} + (2 \beta_{2} - 7 \beta_1) q^{22} + ( - 5 \beta_{2} + 4 \beta_1 - 3) q^{23} + ( - 3 \beta_{2} + 5 \beta_1 + 1) q^{25} + (3 \beta_{2} - \beta_1 + 3) q^{28} + (3 \beta_{2} - \beta_1 + 2) q^{29} + ( - 5 \beta_{2} + 3 \beta_1 - 8) q^{31} + ( - 7 \beta_{2} + 7 \beta_1 - 12) q^{32} + ( - 6 \beta_{2} + 7 \beta_1 - 7) q^{34} + ( - 3 \beta_{2} - \beta_1 - 2) q^{35} + (\beta_1 + 7) q^{37} + ( - 4 \beta_{2} - \beta_1 - 1) q^{38} + ( - 8 \beta_{2} + 7 \beta_1 - 4) q^{40} + ( - 3 \beta_{2} + \beta_1 - 5) q^{41} + ( - 2 \beta_{2} + 4 \beta_1 - 7) q^{43} + ( - 10 \beta_{2} + 7 \beta_1 - 11) q^{44} + (10 \beta_{2} - 12 \beta_1 + 5) q^{46} + ( - \beta_1 - 2) q^{47} + (\beta_{2} + \beta_1 - 5) q^{49} + (14 \beta_{2} - 7 \beta_1 + 8) q^{50} + ( - 3 \beta_{2} - \beta_1 + 5) q^{53} + (8 \beta_{2} + 2 \beta_1 + 3) q^{55} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{56} + ( - 3 \beta_{2} + 6 \beta_1 + 1) q^{58} + ( - 2 \beta_{2} + 4 \beta_1) q^{59} + (6 \beta_{2} - \beta_1 - 2) q^{61} + (3 \beta_{2} - 16 \beta_1 + 7) q^{62} + (7 \beta_{2} - 14 \beta_1 + 7) q^{64} + ( - 7 \beta_{2} + 4 \beta_1) q^{67} + (9 \beta_{2} - 14 \beta_1 + 12) q^{68} + ( - \beta_{2} - 4 \beta_1 - 7) q^{70} + (5 \beta_{2} - 7 \beta_1 + 4) q^{71} + (9 \beta_{2} - 6 \beta_1 + 7) q^{73} + (9 \beta_{2} + 6 \beta_1 - 4) q^{74} + (\beta_{2} - 2 \beta_1 - 6) q^{76} + ( - 2 \beta_{2} - 2 \beta_1 - 5) q^{77} - 3 \beta_{2} q^{79} + (8 \beta_{2} - 5 \beta_1 + 11) q^{80} + ( - 9 \beta_1 + 2) q^{82} + (2 \beta_{2} - \beta_1 - 3) q^{83} + ( - 3 \beta_{2} + \beta_1 + 5) q^{85} + (3 \beta_{2} - 13 \beta_1 + 15) q^{86} + (9 \beta_{2} - 14 \beta_1 + 12) q^{88} + (\beta_{2} + 6 \beta_1 - 2) q^{89} + ( - 19 \beta_{2} + 19 \beta_1 - 15) q^{92} + ( - 4 \beta_{2} - \beta_1 - 1) q^{94} + ( - \beta_{2} + 5 \beta_1 + 5) q^{95} + (2 \beta_{2} - 12 \beta_1 + 3) q^{97} + ( - 4 \beta_{2} - 5 \beta_1 + 10) q^{98}+O(q^{100})$$ q + (b2 + b1 - 1) * q^2 + (-b1 + 4) * q^4 + (b2 - 2*b1 - 1) * q^5 + (b2 + 1) * q^7 + (3*b1 - 5) * q^8 + (-6*b2 + 2*b1 - 3) * q^10 + (-3*b2 + b1 - 3) * q^11 + (2*b1 + 1) * q^14 + (b2 - 6*b1 + 6) * q^16 + (2*b2 - 3*b1 + 2) * q^17 + (-b1 - 2) * q^19 + (5*b2 - 7*b1 - 1) * q^20 + (2*b2 - 7*b1) * q^22 + (-5*b2 + 4*b1 - 3) * q^23 + (-3*b2 + 5*b1 + 1) * q^25 + (3*b2 - b1 + 3) * q^28 + (3*b2 - b1 + 2) * q^29 + (-5*b2 + 3*b1 - 8) * q^31 + (-7*b2 + 7*b1 - 12) * q^32 + (-6*b2 + 7*b1 - 7) * q^34 + (-3*b2 - b1 - 2) * q^35 + (b1 + 7) * q^37 + (-4*b2 - b1 - 1) * q^38 + (-8*b2 + 7*b1 - 4) * q^40 + (-3*b2 + b1 - 5) * q^41 + (-2*b2 + 4*b1 - 7) * q^43 + (-10*b2 + 7*b1 - 11) * q^44 + (10*b2 - 12*b1 + 5) * q^46 + (-b1 - 2) * q^47 + (b2 + b1 - 5) * q^49 + (14*b2 - 7*b1 + 8) * q^50 + (-3*b2 - b1 + 5) * q^53 + (8*b2 + 2*b1 + 3) * q^55 + (-2*b2 + 3*b1 - 2) * q^56 + (-3*b2 + 6*b1 + 1) * q^58 + (-2*b2 + 4*b1) * q^59 + (6*b2 - b1 - 2) * q^61 + (3*b2 - 16*b1 + 7) * q^62 + (7*b2 - 14*b1 + 7) * q^64 + (-7*b2 + 4*b1) * q^67 + (9*b2 - 14*b1 + 12) * q^68 + (-b2 - 4*b1 - 7) * q^70 + (5*b2 - 7*b1 + 4) * q^71 + (9*b2 - 6*b1 + 7) * q^73 + (9*b2 + 6*b1 - 4) * q^74 + (b2 - 2*b1 - 6) * q^76 + (-2*b2 - 2*b1 - 5) * q^77 - 3*b2 * q^79 + (8*b2 - 5*b1 + 11) * q^80 + (-9*b1 + 2) * q^82 + (2*b2 - b1 - 3) * q^83 + (-3*b2 + b1 + 5) * q^85 + (3*b2 - 13*b1 + 15) * q^86 + (9*b2 - 14*b1 + 12) * q^88 + (b2 + 6*b1 - 2) * q^89 + (-19*b2 + 19*b1 - 15) * q^92 + (-4*b2 - b1 - 1) * q^94 + (-b2 + 5*b1 + 5) * q^95 + (2*b2 - 12*b1 + 3) * q^97 + (-4*b2 - 5*b1 + 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 11 * q^4 - 6 * q^5 + 2 * q^7 - 12 * q^8 $$3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8} - q^{10} - 5 q^{11} + 5 q^{14} + 11 q^{16} + q^{17} - 7 q^{19} - 15 q^{20} - 9 q^{22} + 11 q^{25} + 5 q^{28} + 2 q^{29} - 16 q^{31} - 22 q^{32} - 8 q^{34} - 4 q^{35} + 22 q^{37} + 3 q^{40} - 11 q^{41} - 15 q^{43} - 16 q^{44} - 7 q^{46} - 7 q^{47} - 15 q^{49} + 3 q^{50} + 17 q^{53} + 3 q^{55} - q^{56} + 12 q^{58} + 6 q^{59} - 13 q^{61} + 2 q^{62} + 11 q^{67} + 13 q^{68} - 24 q^{70} + 6 q^{73} - 15 q^{74} - 21 q^{76} - 15 q^{77} + 3 q^{79} + 20 q^{80} - 3 q^{82} - 12 q^{83} + 19 q^{85} + 29 q^{86} + 13 q^{88} - q^{89} - 7 q^{92} + 21 q^{95} - 5 q^{97} + 29 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 11 * q^4 - 6 * q^5 + 2 * q^7 - 12 * q^8 - q^10 - 5 * q^11 + 5 * q^14 + 11 * q^16 + q^17 - 7 * q^19 - 15 * q^20 - 9 * q^22 + 11 * q^25 + 5 * q^28 + 2 * q^29 - 16 * q^31 - 22 * q^32 - 8 * q^34 - 4 * q^35 + 22 * q^37 + 3 * q^40 - 11 * q^41 - 15 * q^43 - 16 * q^44 - 7 * q^46 - 7 * q^47 - 15 * q^49 + 3 * q^50 + 17 * q^53 + 3 * q^55 - q^56 + 12 * q^58 + 6 * q^59 - 13 * q^61 + 2 * q^62 + 11 * q^67 + 13 * q^68 - 24 * q^70 + 6 * q^73 - 15 * q^74 - 21 * q^76 - 15 * q^77 + 3 * q^79 + 20 * q^80 - 3 * q^82 - 12 * q^83 + 19 * q^85 + 29 * q^86 + 13 * q^88 - q^89 - 7 * q^92 + 21 * q^95 - 5 * q^97 + 29 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

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
 −1.24698 0.445042 1.80194
−2.69202 0 5.24698 1.04892 0 0.554958 −8.74094 0 −2.82371
1.2 −2.35690 0 3.55496 −3.69202 0 −0.801938 −3.66487 0 8.70171
1.3 2.04892 0 2.19806 −3.35690 0 2.24698 0.405813 0 −6.87800
 $$n$$: 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 1521.2.a.n 3
3.b odd 2 1 507.2.a.l yes 3
12.b even 2 1 8112.2.a.cp 3
13.b even 2 1 1521.2.a.s 3
13.d odd 4 2 1521.2.b.k 6
39.d odd 2 1 507.2.a.i 3
39.f even 4 2 507.2.b.f 6
39.h odd 6 2 507.2.e.l 6
39.i odd 6 2 507.2.e.i 6
39.k even 12 4 507.2.j.i 12
156.h even 2 1 8112.2.a.cg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 39.d odd 2 1
507.2.a.l yes 3 3.b odd 2 1
507.2.b.f 6 39.f even 4 2
507.2.e.i 6 39.i odd 6 2
507.2.e.l 6 39.h odd 6 2
507.2.j.i 12 39.k even 12 4
1521.2.a.n 3 1.a even 1 1 trivial
1521.2.a.s 3 13.b even 2 1
1521.2.b.k 6 13.d odd 4 2
8112.2.a.cg 3 156.h even 2 1
8112.2.a.cp 3 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{3} + 3T_{2}^{2} - 4T_{2} - 13$$ T2^3 + 3*T2^2 - 4*T2 - 13 $$T_{5}^{3} + 6T_{5}^{2} + 5T_{5} - 13$$ T5^3 + 6*T5^2 + 5*T5 - 13 $$T_{7}^{3} - 2T_{7}^{2} - T_{7} + 1$$ T7^3 - 2*T7^2 - T7 + 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3 T^{2} - 4 T - 13$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 6 T^{2} + 5 T - 13$$
$7$ $$T^{3} - 2T^{2} - T + 1$$
$11$ $$T^{3} + 5 T^{2} - 8 T - 41$$
$13$ $$T^{3}$$
$17$ $$T^{3} - T^{2} - 16 T - 13$$
$19$ $$T^{3} + 7 T^{2} + 14 T + 7$$
$23$ $$T^{3} - 49T - 91$$
$29$ $$T^{3} - 2 T^{2} - 15 T + 29$$
$31$ $$T^{3} + 16 T^{2} + 41 T - 197$$
$37$ $$T^{3} - 22 T^{2} + 159 T - 377$$
$41$ $$T^{3} + 11 T^{2} + 24 T - 29$$
$43$ $$T^{3} + 15 T^{2} + 47 T + 41$$
$47$ $$T^{3} + 7 T^{2} + 14 T + 7$$
$53$ $$T^{3} - 17 T^{2} + 66 T + 41$$
$59$ $$T^{3} - 6 T^{2} - 16 T + 104$$
$61$ $$T^{3} + 13 T^{2} - 16 T - 167$$
$67$ $$T^{3} - 11 T^{2} - 46 T - 41$$
$71$ $$T^{3} - 91T - 203$$
$73$ $$T^{3} - 6 T^{2} - 135 T + 923$$
$79$ $$T^{3} - 3 T^{2} - 18 T + 27$$
$83$ $$T^{3} + 12 T^{2} + 41 T + 43$$
$89$ $$T^{3} + T^{2} - 100 T - 113$$
$97$ $$T^{3} + 5 T^{2} - 281 T - 1637$$