# Properties

 Label 507.2.a.l Level $507$ Weight $2$ Character orbit 507.a Self dual yes Analytic conductor $4.048$ Analytic rank $0$ 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: $$0$$ 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: 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1 + 1) q^{2} - q^{3} + ( - \beta_1 + 4) q^{4} + ( - \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + (\beta_{2} + 1) q^{7} + ( - 3 \beta_1 + 5) q^{8} + q^{9}+O(q^{10})$$ q + (-b2 - b1 + 1) * q^2 - q^3 + (-b1 + 4) * q^4 + (-b2 + 2*b1 + 1) * q^5 + (b2 + b1 - 1) * q^6 + (b2 + 1) * q^7 + (-3*b1 + 5) * q^8 + q^9 $$q + ( - \beta_{2} - \beta_1 + 1) q^{2} - q^{3} + ( - \beta_1 + 4) q^{4} + ( - \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + (\beta_{2} + 1) q^{7} + ( - 3 \beta_1 + 5) q^{8} + q^{9} + ( - 6 \beta_{2} + 2 \beta_1 - 3) q^{10} + (3 \beta_{2} - \beta_1 + 3) q^{11} + (\beta_1 - 4) q^{12} + ( - 2 \beta_1 - 1) q^{14} + (\beta_{2} - 2 \beta_1 - 1) q^{15} + (\beta_{2} - 6 \beta_1 + 6) q^{16} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{17} + ( - \beta_{2} - \beta_1 + 1) q^{18} + ( - \beta_1 - 2) q^{19} + ( - 5 \beta_{2} + 7 \beta_1 + 1) q^{20} + ( - \beta_{2} - 1) q^{21} + (2 \beta_{2} - 7 \beta_1) q^{22} + (5 \beta_{2} - 4 \beta_1 + 3) q^{23} + (3 \beta_1 - 5) q^{24} + ( - 3 \beta_{2} + 5 \beta_1 + 1) q^{25} - q^{27} + (3 \beta_{2} - \beta_1 + 3) q^{28} + ( - 3 \beta_{2} + \beta_1 - 2) q^{29} + (6 \beta_{2} - 2 \beta_1 + 3) q^{30} + ( - 5 \beta_{2} + 3 \beta_1 - 8) q^{31} + (7 \beta_{2} - 7 \beta_1 + 12) q^{32} + ( - 3 \beta_{2} + \beta_1 - 3) q^{33} + ( - 6 \beta_{2} + 7 \beta_1 - 7) q^{34} + (3 \beta_{2} + \beta_1 + 2) q^{35} + ( - \beta_1 + 4) q^{36} + (\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_1 + 1) q^{42} + ( - 2 \beta_{2} + 4 \beta_1 - 7) q^{43} + (10 \beta_{2} - 7 \beta_1 + 11) q^{44} + ( - \beta_{2} + 2 \beta_1 + 1) q^{45} + (10 \beta_{2} - 12 \beta_1 + 5) q^{46} + (\beta_1 + 2) q^{47} + ( - \beta_{2} + 6 \beta_1 - 6) q^{48} + (\beta_{2} + \beta_1 - 5) q^{49} + ( - 14 \beta_{2} + 7 \beta_1 - 8) q^{50} + (2 \beta_{2} - 3 \beta_1 + 2) q^{51} + (3 \beta_{2} + \beta_1 - 5) q^{53} + (\beta_{2} + \beta_1 - 1) q^{54} + (8 \beta_{2} + 2 \beta_1 + 3) q^{55} + (2 \beta_{2} - 3 \beta_1 + 2) q^{56} + (\beta_1 + 2) q^{57} + ( - 3 \beta_{2} + 6 \beta_1 + 1) q^{58} + (2 \beta_{2} - 4 \beta_1) q^{59} + (5 \beta_{2} - 7 \beta_1 - 1) q^{60} + (6 \beta_{2} - \beta_1 - 2) q^{61} + ( - 3 \beta_{2} + 16 \beta_1 - 7) q^{62} + (\beta_{2} + 1) q^{63} + (7 \beta_{2} - 14 \beta_1 + 7) q^{64} + ( - 2 \beta_{2} + 7 \beta_1) q^{66} + ( - 7 \beta_{2} + 4 \beta_1) q^{67} + ( - 9 \beta_{2} + 14 \beta_1 - 12) q^{68} + ( - 5 \beta_{2} + 4 \beta_1 - 3) q^{69} + ( - \beta_{2} - 4 \beta_1 - 7) q^{70} + ( - 5 \beta_{2} + 7 \beta_1 - 4) q^{71} + ( - 3 \beta_1 + 5) q^{72} + (9 \beta_{2} - 6 \beta_1 + 7) q^{73} + ( - 9 \beta_{2} - 6 \beta_1 + 4) q^{74} + (3 \beta_{2} - 5 \beta_1 - 1) q^{75} + (\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} + q^{81} + ( - 9 \beta_1 + 2) q^{82} + ( - 2 \beta_{2} + \beta_1 + 3) q^{83} + ( - 3 \beta_{2} + \beta_1 - 3) q^{84} + ( - 3 \beta_{2} + \beta_1 + 5) q^{85} + ( - 3 \beta_{2} + 13 \beta_1 - 15) q^{86} + (3 \beta_{2} - \beta_1 + 2) q^{87} + (9 \beta_{2} - 14 \beta_1 + 12) q^{88} + ( - \beta_{2} - 6 \beta_1 + 2) q^{89} + ( - 6 \beta_{2} + 2 \beta_1 - 3) q^{90} + (19 \beta_{2} - 19 \beta_1 + 15) q^{92} + (5 \beta_{2} - 3 \beta_1 + 8) q^{93} + ( - 4 \beta_{2} - \beta_1 - 1) q^{94} + (\beta_{2} - 5 \beta_1 - 5) q^{95} + ( - 7 \beta_{2} + 7 \beta_1 - 12) q^{96} + (2 \beta_{2} - 12 \beta_1 + 3) q^{97} + (4 \beta_{2} + 5 \beta_1 - 10) q^{98} + (3 \beta_{2} - \beta_1 + 3) q^{99}+O(q^{100})$$ q + (-b2 - b1 + 1) * q^2 - q^3 + (-b1 + 4) * q^4 + (-b2 + 2*b1 + 1) * q^5 + (b2 + b1 - 1) * q^6 + (b2 + 1) * q^7 + (-3*b1 + 5) * q^8 + q^9 + (-6*b2 + 2*b1 - 3) * q^10 + (3*b2 - b1 + 3) * q^11 + (b1 - 4) * q^12 + (-2*b1 - 1) * q^14 + (b2 - 2*b1 - 1) * q^15 + (b2 - 6*b1 + 6) * q^16 + (-2*b2 + 3*b1 - 2) * q^17 + (-b2 - b1 + 1) * q^18 + (-b1 - 2) * q^19 + (-5*b2 + 7*b1 + 1) * q^20 + (-b2 - 1) * q^21 + (2*b2 - 7*b1) * q^22 + (5*b2 - 4*b1 + 3) * q^23 + (3*b1 - 5) * q^24 + (-3*b2 + 5*b1 + 1) * q^25 - q^27 + (3*b2 - b1 + 3) * q^28 + (-3*b2 + b1 - 2) * q^29 + (6*b2 - 2*b1 + 3) * q^30 + (-5*b2 + 3*b1 - 8) * q^31 + (7*b2 - 7*b1 + 12) * q^32 + (-3*b2 + b1 - 3) * q^33 + (-6*b2 + 7*b1 - 7) * q^34 + (3*b2 + b1 + 2) * q^35 + (-b1 + 4) * q^36 + (b1 + 7) * q^37 + (4*b2 + b1 + 1) * q^38 + (-8*b2 + 7*b1 - 4) * q^40 + (3*b2 - b1 + 5) * q^41 + (2*b1 + 1) * q^42 + (-2*b2 + 4*b1 - 7) * q^43 + (10*b2 - 7*b1 + 11) * q^44 + (-b2 + 2*b1 + 1) * q^45 + (10*b2 - 12*b1 + 5) * q^46 + (b1 + 2) * q^47 + (-b2 + 6*b1 - 6) * q^48 + (b2 + b1 - 5) * q^49 + (-14*b2 + 7*b1 - 8) * q^50 + (2*b2 - 3*b1 + 2) * q^51 + (3*b2 + b1 - 5) * q^53 + (b2 + b1 - 1) * q^54 + (8*b2 + 2*b1 + 3) * q^55 + (2*b2 - 3*b1 + 2) * q^56 + (b1 + 2) * q^57 + (-3*b2 + 6*b1 + 1) * q^58 + (2*b2 - 4*b1) * q^59 + (5*b2 - 7*b1 - 1) * q^60 + (6*b2 - b1 - 2) * q^61 + (-3*b2 + 16*b1 - 7) * q^62 + (b2 + 1) * q^63 + (7*b2 - 14*b1 + 7) * q^64 + (-2*b2 + 7*b1) * q^66 + (-7*b2 + 4*b1) * q^67 + (-9*b2 + 14*b1 - 12) * q^68 + (-5*b2 + 4*b1 - 3) * q^69 + (-b2 - 4*b1 - 7) * q^70 + (-5*b2 + 7*b1 - 4) * q^71 + (-3*b1 + 5) * q^72 + (9*b2 - 6*b1 + 7) * q^73 + (-9*b2 - 6*b1 + 4) * q^74 + (3*b2 - 5*b1 - 1) * q^75 + (b2 - 2*b1 - 6) * q^76 + (2*b2 + 2*b1 + 5) * q^77 - 3*b2 * q^79 + (-8*b2 + 5*b1 - 11) * q^80 + q^81 + (-9*b1 + 2) * q^82 + (-2*b2 + b1 + 3) * q^83 + (-3*b2 + b1 - 3) * q^84 + (-3*b2 + b1 + 5) * q^85 + (-3*b2 + 13*b1 - 15) * q^86 + (3*b2 - b1 + 2) * q^87 + (9*b2 - 14*b1 + 12) * q^88 + (-b2 - 6*b1 + 2) * q^89 + (-6*b2 + 2*b1 - 3) * q^90 + (19*b2 - 19*b1 + 15) * q^92 + (5*b2 - 3*b1 + 8) * q^93 + (-4*b2 - b1 - 1) * q^94 + (b2 - 5*b1 - 5) * q^95 + (-7*b2 + 7*b1 - 12) * q^96 + (2*b2 - 12*b1 + 3) * q^97 + (4*b2 + 5*b1 - 10) * q^98 + (3*b2 - b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 11 q^{4} + 6 q^{5} - 3 q^{6} + 2 q^{7} + 12 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^3 + 11 * q^4 + 6 * q^5 - 3 * q^6 + 2 * q^7 + 12 * q^8 + 3 * q^9 $$3 q + 3 q^{2} - 3 q^{3} + 11 q^{4} + 6 q^{5} - 3 q^{6} + 2 q^{7} + 12 q^{8} + 3 q^{9} - q^{10} + 5 q^{11} - 11 q^{12} - 5 q^{14} - 6 q^{15} + 11 q^{16} - q^{17} + 3 q^{18} - 7 q^{19} + 15 q^{20} - 2 q^{21} - 9 q^{22} - 12 q^{24} + 11 q^{25} - 3 q^{27} + 5 q^{28} - 2 q^{29} + q^{30} - 16 q^{31} + 22 q^{32} - 5 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} + 22 q^{37} + 3 q^{40} + 11 q^{41} + 5 q^{42} - 15 q^{43} + 16 q^{44} + 6 q^{45} - 7 q^{46} + 7 q^{47} - 11 q^{48} - 15 q^{49} - 3 q^{50} + q^{51} - 17 q^{53} - 3 q^{54} + 3 q^{55} + q^{56} + 7 q^{57} + 12 q^{58} - 6 q^{59} - 15 q^{60} - 13 q^{61} - 2 q^{62} + 2 q^{63} + 9 q^{66} + 11 q^{67} - 13 q^{68} - 24 q^{70} + 12 q^{72} + 6 q^{73} + 15 q^{74} - 11 q^{75} - 21 q^{76} + 15 q^{77} + 3 q^{79} - 20 q^{80} + 3 q^{81} - 3 q^{82} + 12 q^{83} - 5 q^{84} + 19 q^{85} - 29 q^{86} + 2 q^{87} + 13 q^{88} + q^{89} - q^{90} + 7 q^{92} + 16 q^{93} - 21 q^{95} - 22 q^{96} - 5 q^{97} - 29 q^{98} + 5 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^3 + 11 * q^4 + 6 * q^5 - 3 * q^6 + 2 * q^7 + 12 * q^8 + 3 * q^9 - q^10 + 5 * q^11 - 11 * q^12 - 5 * q^14 - 6 * q^15 + 11 * q^16 - q^17 + 3 * q^18 - 7 * q^19 + 15 * q^20 - 2 * q^21 - 9 * q^22 - 12 * q^24 + 11 * q^25 - 3 * q^27 + 5 * q^28 - 2 * q^29 + q^30 - 16 * q^31 + 22 * q^32 - 5 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 + 22 * q^37 + 3 * q^40 + 11 * q^41 + 5 * q^42 - 15 * q^43 + 16 * q^44 + 6 * q^45 - 7 * q^46 + 7 * q^47 - 11 * q^48 - 15 * q^49 - 3 * q^50 + q^51 - 17 * q^53 - 3 * q^54 + 3 * q^55 + q^56 + 7 * q^57 + 12 * q^58 - 6 * q^59 - 15 * q^60 - 13 * q^61 - 2 * q^62 + 2 * q^63 + 9 * q^66 + 11 * q^67 - 13 * q^68 - 24 * q^70 + 12 * q^72 + 6 * q^73 + 15 * q^74 - 11 * q^75 - 21 * q^76 + 15 * q^77 + 3 * q^79 - 20 * q^80 + 3 * q^81 - 3 * q^82 + 12 * q^83 - 5 * q^84 + 19 * q^85 - 29 * q^86 + 2 * q^87 + 13 * q^88 + q^89 - q^90 + 7 * q^92 + 16 * q^93 - 21 * q^95 - 22 * q^96 - 5 * q^97 - 29 * q^98 + 5 * q^99

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
−2.04892 −1.00000 2.19806 3.35690 2.04892 2.24698 −0.405813 1.00000 −6.87800
1.2 2.35690 −1.00000 3.55496 3.69202 −2.35690 −0.801938 3.66487 1.00000 8.70171
1.3 2.69202 −1.00000 5.24698 −1.04892 −2.69202 0.554958 8.74094 1.00000 −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 507.2.a.l yes 3
3.b odd 2 1 1521.2.a.n 3
4.b odd 2 1 8112.2.a.cp 3
13.b even 2 1 507.2.a.i 3
13.c even 3 2 507.2.e.i 6
13.d odd 4 2 507.2.b.f 6
13.e even 6 2 507.2.e.l 6
13.f odd 12 4 507.2.j.i 12
39.d odd 2 1 1521.2.a.s 3
39.f even 4 2 1521.2.b.k 6
52.b odd 2 1 8112.2.a.cg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 13.b even 2 1
507.2.a.l yes 3 1.a even 1 1 trivial
507.2.b.f 6 13.d odd 4 2
507.2.e.i 6 13.c even 3 2
507.2.e.l 6 13.e even 6 2
507.2.j.i 12 13.f odd 12 4
1521.2.a.n 3 3.b odd 2 1
1521.2.a.s 3 39.d odd 2 1
1521.2.b.k 6 39.f even 4 2
8112.2.a.cg 3 52.b odd 2 1
8112.2.a.cp 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

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

## Hecke characteristic polynomials

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