# Properties

 Label 1521.2.a.s 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: $$x^{3} - x^{2} - 2 x + 1$$ 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.

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

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 3.b odd 2 1
507.2.a.l yes 3 39.d odd 2 1
507.2.b.f 6 39.f even 4 2
507.2.e.i 6 39.h odd 6 2
507.2.e.l 6 39.i odd 6 2
507.2.j.i 12 39.k even 12 4
1521.2.a.n 3 13.b even 2 1
1521.2.a.s 3 1.a even 1 1 trivial
1521.2.b.k 6 13.d odd 4 2
8112.2.a.cg 3 12.b even 2 1
8112.2.a.cp 3 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{3} - 3 T_{2}^{2} - 4 T_{2} + 13$$ $$T_{5}^{3} - 6 T_{5}^{2} + 5 T_{5} + 13$$ $$T_{7}^{3} + 2 T_{7}^{2} - T_{7} - 1$$

## Hecke characteristic polynomials

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