# Properties

 Label 1521.2.a.p Level $1521$ Weight $2$ Character orbit 1521.a Self dual yes Analytic conductor $12.145$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ 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

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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} + (\beta_{2} - \beta_1 + 4) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + b2 * q^4 + (-b2 + b1 - 2) * q^5 + (b2 - b1 + 4) * q^7 + (-b2 + 2*b1 - 1) * q^8 $$q - \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} + (\beta_{2} - \beta_1 + 4) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8} + (2 \beta_1 - 1) q^{10} + (4 \beta_{2} - \beta_1 + 2) q^{11} + ( - 4 \beta_1 + 1) q^{14} + ( - 3 \beta_{2} + \beta_1 - 3) q^{16} + ( - \beta_{2} + 2) q^{17} + ( - \beta_{2} + 4 \beta_1 + 2) q^{19} - \beta_1 q^{20} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{22} + (5 \beta_{2} - 3 \beta_1 + 2) q^{23} + (2 \beta_{2} - 3 \beta_1) q^{25} + (2 \beta_{2} + \beta_1) q^{28} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{29} + ( - 2 \beta_{2} - 3 \beta_1 + 3) q^{31} + (4 \beta_{2} - \beta_1 + 3) q^{32} + (\beta_{2} - 2 \beta_1 + 1) q^{34} + ( - 4 \beta_{2} + 5 \beta_1 - 9) q^{35} + (\beta_{2} + 5) q^{37} + ( - 3 \beta_{2} - 2 \beta_1 - 7) q^{38} + (\beta_{2} - 4 \beta_1 + 4) q^{40} - \beta_1 q^{41} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{43} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{44} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{46} + ( - 3 \beta_{2} - 6 \beta_1 + 4) q^{47} + (6 \beta_{2} - 7 \beta_1 + 10) q^{49} + (\beta_{2} + 4) q^{50} + (2 \beta_{2} - 3 \beta_1 + 6) q^{53} + ( - 2 \beta_{2} - 5) q^{55} + ( - 3 \beta_{2} + 8 \beta_1 - 6) q^{56} + ( - \beta_{2} - \beta_1 - 4) q^{58} + ( - 6 \beta_{2} + 2 \beta_1 + 2) q^{59} + (3 \beta_{2} + 2 \beta_1 - 4) q^{61} + (5 \beta_{2} - 3 \beta_1 + 8) q^{62} + (3 \beta_{2} - 5 \beta_1 + 4) q^{64} + ( - 3 \beta_{2} - \beta_1 + 1) q^{67} + (3 \beta_{2} - \beta_1 - 1) q^{68} + ( - \beta_{2} + 9 \beta_1 - 6) q^{70} + ( - 2 \beta_{2} + 7 \beta_1 - 1) q^{71} + ( - \beta_{2} - \beta_1 + 6) q^{73} + ( - \beta_{2} - 5 \beta_1 - 1) q^{74} + (7 \beta_{2} - \beta_1 + 3) q^{76} + (10 \beta_{2} - 2 \beta_1 + 9) q^{77} + ( - 5 \beta_{2} + \beta_1 - 5) q^{79} + (3 \beta_{2} - 2 \beta_1 + 7) q^{80} + (\beta_{2} + 2) q^{82} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{83} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{85} + (4 \beta_{2} + \beta_1 + 6) q^{86} + (5 \beta_{2} + \beta_1 - 1) q^{88} + (\beta_{2} - 3 \beta_1 + 3) q^{89} + ( - 6 \beta_{2} + 5 \beta_1 + 2) q^{92} + (9 \beta_{2} - 4 \beta_1 + 15) q^{94} + ( - 2 \beta_{2} - 5 \beta_1) q^{95} + (10 \beta_{2} - 6 \beta_1 + 7) q^{97} + (\beta_{2} - 10 \beta_1 + 8) q^{98}+O(q^{100})$$ q - b1 * q^2 + b2 * q^4 + (-b2 + b1 - 2) * q^5 + (b2 - b1 + 4) * q^7 + (-b2 + 2*b1 - 1) * q^8 + (2*b1 - 1) * q^10 + (4*b2 - b1 + 2) * q^11 + (-4*b1 + 1) * q^14 + (-3*b2 + b1 - 3) * q^16 + (-b2 + 2) * q^17 + (-b2 + 4*b1 + 2) * q^19 - b1 * q^20 + (-3*b2 - 2*b1 - 2) * q^22 + (5*b2 - 3*b1 + 2) * q^23 + (2*b2 - 3*b1) * q^25 + (2*b2 + b1) * q^28 + (-2*b2 + 3*b1 + 1) * q^29 + (-2*b2 - 3*b1 + 3) * q^31 + (4*b2 - b1 + 3) * q^32 + (b2 - 2*b1 + 1) * q^34 + (-4*b2 + 5*b1 - 9) * q^35 + (b2 + 5) * q^37 + (-3*b2 - 2*b1 - 7) * q^38 + (b2 - 4*b1 + 4) * q^40 - b1 * q^41 + (-2*b2 - 2*b1 - 1) * q^43 + (-3*b2 + 4*b1 + 3) * q^44 + (-2*b2 - 2*b1 + 1) * q^46 + (-3*b2 - 6*b1 + 4) * q^47 + (6*b2 - 7*b1 + 10) * q^49 + (b2 + 4) * q^50 + (2*b2 - 3*b1 + 6) * q^53 + (-2*b2 - 5) * q^55 + (-3*b2 + 8*b1 - 6) * q^56 + (-b2 - b1 - 4) * q^58 + (-6*b2 + 2*b1 + 2) * q^59 + (3*b2 + 2*b1 - 4) * q^61 + (5*b2 - 3*b1 + 8) * q^62 + (3*b2 - 5*b1 + 4) * q^64 + (-3*b2 - b1 + 1) * q^67 + (3*b2 - b1 - 1) * q^68 + (-b2 + 9*b1 - 6) * q^70 + (-2*b2 + 7*b1 - 1) * q^71 + (-b2 - b1 + 6) * q^73 + (-b2 - 5*b1 - 1) * q^74 + (7*b2 - b1 + 3) * q^76 + (10*b2 - 2*b1 + 9) * q^77 + (-5*b2 + b1 - 5) * q^79 + (3*b2 - 2*b1 + 7) * q^80 + (b2 + 2) * q^82 + (-3*b2 + 4*b1 + 3) * q^83 + (-2*b2 + 3*b1 - 4) * q^85 + (4*b2 + b1 + 6) * q^86 + (5*b2 + b1 - 1) * q^88 + (b2 - 3*b1 + 3) * q^89 + (-6*b2 + 5*b1 + 2) * q^92 + (9*b2 - 4*b1 + 15) * q^94 + (-2*b2 - 5*b1) * q^95 + (10*b2 - 6*b1 + 7) * q^97 + (b2 - 10*b1 + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - q^{4} - 4 q^{5} + 10 q^{7}+O(q^{10})$$ 3 * q - q^2 - q^4 - 4 * q^5 + 10 * q^7 $$3 q - q^{2} - q^{4} - 4 q^{5} + 10 q^{7} - q^{10} + q^{11} - q^{14} - 5 q^{16} + 7 q^{17} + 11 q^{19} - q^{20} - 5 q^{22} - 2 q^{23} - 5 q^{25} - q^{28} + 8 q^{29} + 8 q^{31} + 4 q^{32} - 18 q^{35} + 14 q^{37} - 20 q^{38} + 7 q^{40} - q^{41} - 3 q^{43} + 16 q^{44} + 3 q^{46} + 9 q^{47} + 17 q^{49} + 11 q^{50} + 13 q^{53} - 13 q^{55} - 7 q^{56} - 12 q^{58} + 14 q^{59} - 13 q^{61} + 16 q^{62} + 4 q^{64} + 5 q^{67} - 7 q^{68} - 8 q^{70} + 6 q^{71} + 18 q^{73} - 7 q^{74} + q^{76} + 15 q^{77} - 9 q^{79} + 16 q^{80} + 5 q^{82} + 16 q^{83} - 7 q^{85} + 15 q^{86} - 7 q^{88} + 5 q^{89} + 17 q^{92} + 32 q^{94} - 3 q^{95} + 5 q^{97} + 13 q^{98}+O(q^{100})$$ 3 * q - q^2 - q^4 - 4 * q^5 + 10 * q^7 - q^10 + q^11 - q^14 - 5 * q^16 + 7 * q^17 + 11 * q^19 - q^20 - 5 * q^22 - 2 * q^23 - 5 * q^25 - q^28 + 8 * q^29 + 8 * q^31 + 4 * q^32 - 18 * q^35 + 14 * q^37 - 20 * q^38 + 7 * q^40 - q^41 - 3 * q^43 + 16 * q^44 + 3 * q^46 + 9 * q^47 + 17 * q^49 + 11 * q^50 + 13 * q^53 - 13 * q^55 - 7 * q^56 - 12 * q^58 + 14 * q^59 - 13 * q^61 + 16 * q^62 + 4 * q^64 + 5 * q^67 - 7 * q^68 - 8 * q^70 + 6 * q^71 + 18 * q^73 - 7 * q^74 + q^76 + 15 * q^77 - 9 * q^79 + 16 * q^80 + 5 * q^82 + 16 * q^83 - 7 * q^85 + 15 * q^86 - 7 * q^88 + 5 * q^89 + 17 * q^92 + 32 * q^94 - 3 * q^95 + 5 * q^97 + 13 * 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.80194 0.445042 −1.24698
−1.80194 0 1.24698 −1.44504 0 3.44504 1.35690 0 2.60388
1.2 −0.445042 0 −1.80194 0.246980 0 1.75302 1.69202 0 −0.109916
1.3 1.24698 0 −0.445042 −2.80194 0 4.80194 −3.04892 0 −3.49396
 $$n$$: 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.p 3
3.b odd 2 1 507.2.a.k yes 3
12.b even 2 1 8112.2.a.cf 3
13.b even 2 1 1521.2.a.q 3
13.d odd 4 2 1521.2.b.m 6
39.d odd 2 1 507.2.a.j 3
39.f even 4 2 507.2.b.g 6
39.h odd 6 2 507.2.e.k 6
39.i odd 6 2 507.2.e.j 6
39.k even 12 4 507.2.j.h 12
156.h even 2 1 8112.2.a.by 3

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

 $$T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1$$ T2^3 + T2^2 - 2*T2 - 1 $$T_{5}^{3} + 4T_{5}^{2} + 3T_{5} - 1$$ T5^3 + 4*T5^2 + 3*T5 - 1 $$T_{7}^{3} - 10T_{7}^{2} + 31T_{7} - 29$$ T7^3 - 10*T7^2 + 31*T7 - 29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 2T - 1$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 4 T^{2} + 3 T - 1$$
$7$ $$T^{3} - 10 T^{2} + 31 T - 29$$
$11$ $$T^{3} - T^{2} - 30 T + 43$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 7 T^{2} + 14 T - 7$$
$19$ $$T^{3} - 11 T^{2} + 10 T + 113$$
$23$ $$T^{3} + 2 T^{2} - 43 T + 83$$
$29$ $$T^{3} - 8 T^{2} + 5 T + 43$$
$31$ $$T^{3} - 8 T^{2} - 23 T + 197$$
$37$ $$T^{3} - 14 T^{2} + 63 T - 91$$
$41$ $$T^{3} + T^{2} - 2T - 1$$
$43$ $$T^{3} + 3 T^{2} - 25 T + 29$$
$47$ $$T^{3} - 9 T^{2} - 120 T + 911$$
$53$ $$T^{3} - 13 T^{2} + 40 T - 29$$
$59$ $$T^{3} - 14T^{2} + 56$$
$61$ $$T^{3} + 13 T^{2} + 12 T - 223$$
$67$ $$T^{3} - 5 T^{2} - 22 T + 97$$
$71$ $$T^{3} - 6 T^{2} - 79 T + 461$$
$73$ $$T^{3} - 18 T^{2} + 101 T - 167$$
$79$ $$T^{3} + 9 T^{2} - 22 T - 169$$
$83$ $$T^{3} - 16 T^{2} + 55 T + 43$$
$89$ $$T^{3} - 5 T^{2} - 8 T - 1$$
$97$ $$T^{3} - 5 T^{2} - 169 T + 1189$$