# Properties

 Label 507.2.a.i Level $507$ Weight $2$ Character orbit 507.a Self dual yes Analytic conductor $4.048$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.04841538248$$ Analytic rank: $$1$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{2} ) q^{2} - q^{3} + ( 4 - \beta_{1} ) q^{4} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} + ( -1 - \beta_{2} ) q^{7} + ( -5 + 3 \beta_{1} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{2} ) q^{2} - q^{3} + ( 4 - \beta_{1} ) q^{4} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} + ( -1 - \beta_{2} ) q^{7} + ( -5 + 3 \beta_{1} ) q^{8} + q^{9} + ( -3 + 2 \beta_{1} - 6 \beta_{2} ) q^{10} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{11} + ( -4 + \beta_{1} ) q^{12} + ( -1 - 2 \beta_{1} ) q^{14} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{15} + ( 6 - 6 \beta_{1} + \beta_{2} ) q^{16} + ( -2 + 3 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} ) q^{18} + ( 2 + \beta_{1} ) q^{19} + ( -1 - 7 \beta_{1} + 5 \beta_{2} ) q^{20} + ( 1 + \beta_{2} ) q^{21} + ( -7 \beta_{1} + 2 \beta_{2} ) q^{22} + ( 3 - 4 \beta_{1} + 5 \beta_{2} ) q^{23} + ( 5 - 3 \beta_{1} ) q^{24} + ( 1 + 5 \beta_{1} - 3 \beta_{2} ) q^{25} - q^{27} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{28} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{29} + ( 3 - 2 \beta_{1} + 6 \beta_{2} ) q^{30} + ( 8 - 3 \beta_{1} + 5 \beta_{2} ) q^{31} + ( -12 + 7 \beta_{1} - 7 \beta_{2} ) q^{32} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{33} + ( 7 - 7 \beta_{1} + 6 \beta_{2} ) q^{34} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{35} + ( 4 - \beta_{1} ) q^{36} + ( -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} + ( 1 + 2 \beta_{1} ) q^{42} + ( -7 + 4 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -11 + 7 \beta_{1} - 10 \beta_{2} ) q^{44} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{45} + ( -5 + 12 \beta_{1} - 10 \beta_{2} ) q^{46} + ( -2 - \beta_{1} ) q^{47} + ( -6 + 6 \beta_{1} - \beta_{2} ) q^{48} + ( -5 + \beta_{1} + \beta_{2} ) q^{49} + ( 8 - 7 \beta_{1} + 14 \beta_{2} ) q^{50} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{51} + ( -5 + \beta_{1} + 3 \beta_{2} ) q^{53} + ( 1 - \beta_{1} - \beta_{2} ) q^{54} + ( 3 + 2 \beta_{1} + 8 \beta_{2} ) q^{55} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{56} + ( -2 - \beta_{1} ) q^{57} + ( -1 - 6 \beta_{1} + 3 \beta_{2} ) q^{58} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 1 + 7 \beta_{1} - 5 \beta_{2} ) q^{60} + ( -2 - \beta_{1} + 6 \beta_{2} ) q^{61} + ( -7 + 16 \beta_{1} - 3 \beta_{2} ) q^{62} + ( -1 - \beta_{2} ) q^{63} + ( 7 - 14 \beta_{1} + 7 \beta_{2} ) q^{64} + ( 7 \beta_{1} - 2 \beta_{2} ) q^{66} + ( -4 \beta_{1} + 7 \beta_{2} ) q^{67} + ( -12 + 14 \beta_{1} - 9 \beta_{2} ) q^{68} + ( -3 + 4 \beta_{1} - 5 \beta_{2} ) q^{69} + ( 7 + 4 \beta_{1} + \beta_{2} ) q^{70} + ( 4 - 7 \beta_{1} + 5 \beta_{2} ) q^{71} + ( -5 + 3 \beta_{1} ) q^{72} + ( -7 + 6 \beta_{1} - 9 \beta_{2} ) q^{73} + ( 4 - 6 \beta_{1} - 9 \beta_{2} ) q^{74} + ( -1 - 5 \beta_{1} + 3 \beta_{2} ) q^{75} + ( 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} + q^{81} + ( 2 - 9 \beta_{1} ) q^{82} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{83} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{84} + ( -5 - \beta_{1} + 3 \beta_{2} ) q^{85} + ( 15 - 13 \beta_{1} + 3 \beta_{2} ) q^{86} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{87} + ( 12 - 14 \beta_{1} + 9 \beta_{2} ) q^{88} + ( -2 + 6 \beta_{1} + \beta_{2} ) q^{89} + ( -3 + 2 \beta_{1} - 6 \beta_{2} ) q^{90} + ( 15 - 19 \beta_{1} + 19 \beta_{2} ) q^{92} + ( -8 + 3 \beta_{1} - 5 \beta_{2} ) q^{93} + ( -1 - \beta_{1} - 4 \beta_{2} ) q^{94} + ( -5 - 5 \beta_{1} + \beta_{2} ) q^{95} + ( 12 - 7 \beta_{1} + 7 \beta_{2} ) q^{96} + ( -3 + 12 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 10 - 5 \beta_{1} - 4 \beta_{2} ) q^{98} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} - 3q^{3} + 11q^{4} - 6q^{5} + 3q^{6} - 2q^{7} - 12q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{2} - 3q^{3} + 11q^{4} - 6q^{5} + 3q^{6} - 2q^{7} - 12q^{8} + 3q^{9} - q^{10} - 5q^{11} - 11q^{12} - 5q^{14} + 6q^{15} + 11q^{16} - q^{17} - 3q^{18} + 7q^{19} - 15q^{20} + 2q^{21} - 9q^{22} + 12q^{24} + 11q^{25} - 3q^{27} - 5q^{28} - 2q^{29} + q^{30} + 16q^{31} - 22q^{32} + 5q^{33} + 8q^{34} + 4q^{35} + 11q^{36} - 22q^{37} + 3q^{40} - 11q^{41} + 5q^{42} - 15q^{43} - 16q^{44} - 6q^{45} + 7q^{46} - 7q^{47} - 11q^{48} - 15q^{49} + 3q^{50} + q^{51} - 17q^{53} + 3q^{54} + 3q^{55} + q^{56} - 7q^{57} - 12q^{58} + 6q^{59} + 15q^{60} - 13q^{61} - 2q^{62} - 2q^{63} + 9q^{66} - 11q^{67} - 13q^{68} + 24q^{70} - 12q^{72} - 6q^{73} + 15q^{74} - 11q^{75} + 21q^{76} + 15q^{77} + 3q^{79} + 20q^{80} + 3q^{81} - 3q^{82} - 12q^{83} + 5q^{84} - 19q^{85} + 29q^{86} + 2q^{87} + 13q^{88} - q^{89} - q^{90} + 7q^{92} - 16q^{93} - 21q^{95} + 22q^{96} + 5q^{97} + 29q^{98} - 5q^{99} + O(q^{100})$$

## 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
 −1.24698 0.445042 1.80194
−2.69202 −1.00000 5.24698 1.04892 2.69202 −0.554958 −8.74094 1.00000 −2.82371
1.2 −2.35690 −1.00000 3.55496 −3.69202 2.35690 0.801938 −3.66487 1.00000 8.70171
1.3 2.04892 −1.00000 2.19806 −3.35690 −2.04892 −2.24698 0.405813 1.00000 −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 507.2.a.i 3
3.b odd 2 1 1521.2.a.s 3
4.b odd 2 1 8112.2.a.cg 3
13.b even 2 1 507.2.a.l yes 3
13.c even 3 2 507.2.e.l 6
13.d odd 4 2 507.2.b.f 6
13.e even 6 2 507.2.e.i 6
13.f odd 12 4 507.2.j.i 12
39.d odd 2 1 1521.2.a.n 3
39.f even 4 2 1521.2.b.k 6
52.b odd 2 1 8112.2.a.cp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 1.a even 1 1 trivial
507.2.a.l yes 3 13.b even 2 1
507.2.b.f 6 13.d odd 4 2
507.2.e.i 6 13.e even 6 2
507.2.e.l 6 13.c even 3 2
507.2.j.i 12 13.f odd 12 4
1521.2.a.n 3 39.d odd 2 1
1521.2.a.s 3 3.b odd 2 1
1521.2.b.k 6 39.f even 4 2
8112.2.a.cg 3 4.b odd 2 1
8112.2.a.cp 3 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

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

## Hecke characteristic polynomials

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