# Properties

 Label 507.2.a.g Level $507$ Weight $2$ Character orbit 507.a Self dual yes Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + ( 2 + \beta ) q^{4} + ( -2 + \beta ) q^{5} + \beta q^{6} + ( -1 - \beta ) q^{7} + ( 4 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} + q^{3} + ( 2 + \beta ) q^{4} + ( -2 + \beta ) q^{5} + \beta q^{6} + ( -1 - \beta ) q^{7} + ( 4 + \beta ) q^{8} + q^{9} + ( 4 - \beta ) q^{10} -2 q^{11} + ( 2 + \beta ) q^{12} + ( -4 - 2 \beta ) q^{14} + ( -2 + \beta ) q^{15} + 3 \beta q^{16} + \beta q^{17} + \beta q^{18} + ( 4 - 2 \beta ) q^{19} + \beta q^{20} + ( -1 - \beta ) q^{21} -2 \beta q^{22} + 2 q^{23} + ( 4 + \beta ) q^{24} + ( 3 - 3 \beta ) q^{25} + q^{27} + ( -6 - 4 \beta ) q^{28} + ( 2 - 3 \beta ) q^{29} + ( 4 - \beta ) q^{30} + ( 1 - \beta ) q^{31} + ( 4 + \beta ) q^{32} -2 q^{33} + ( 4 + \beta ) q^{34} -2 q^{35} + ( 2 + \beta ) q^{36} + ( 6 - \beta ) q^{37} + ( -8 + 2 \beta ) q^{38} + ( -4 + 3 \beta ) q^{40} + \beta q^{41} + ( -4 - 2 \beta ) q^{42} + ( 3 - \beta ) q^{43} + ( -4 - 2 \beta ) q^{44} + ( -2 + \beta ) q^{45} + 2 \beta q^{46} + ( 2 - 4 \beta ) q^{47} + 3 \beta q^{48} + ( -2 + 3 \beta ) q^{49} -12 q^{50} + \beta q^{51} + ( 4 + 3 \beta ) q^{53} + \beta q^{54} + ( 4 - 2 \beta ) q^{55} + ( -8 - 6 \beta ) q^{56} + ( 4 - 2 \beta ) q^{57} + ( -12 - \beta ) q^{58} + ( -6 - 2 \beta ) q^{59} + \beta q^{60} + ( 7 + 2 \beta ) q^{61} -4 q^{62} + ( -1 - \beta ) q^{63} + ( 4 - \beta ) q^{64} -2 \beta q^{66} + ( 3 - \beta ) q^{67} + ( 4 + 3 \beta ) q^{68} + 2 q^{69} -2 \beta q^{70} + 14 q^{71} + ( 4 + \beta ) q^{72} + ( -7 + 2 \beta ) q^{73} + ( -4 + 5 \beta ) q^{74} + ( 3 - 3 \beta ) q^{75} -2 \beta q^{76} + ( 2 + 2 \beta ) q^{77} + ( 7 + \beta ) q^{79} + ( 12 - 3 \beta ) q^{80} + q^{81} + ( 4 + \beta ) q^{82} + ( -4 - 2 \beta ) q^{83} + ( -6 - 4 \beta ) q^{84} + ( 4 - \beta ) q^{85} + ( -4 + 2 \beta ) q^{86} + ( 2 - 3 \beta ) q^{87} + ( -8 - 2 \beta ) q^{88} + ( 8 + 2 \beta ) q^{89} + ( 4 - \beta ) q^{90} + ( 4 + 2 \beta ) q^{92} + ( 1 - \beta ) q^{93} + ( -16 - 2 \beta ) q^{94} + ( -16 + 6 \beta ) q^{95} + ( 4 + \beta ) q^{96} + ( -7 + \beta ) q^{97} + ( 12 + \beta ) q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 2q^{3} + 5q^{4} - 3q^{5} + q^{6} - 3q^{7} + 9q^{8} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} + 2q^{3} + 5q^{4} - 3q^{5} + q^{6} - 3q^{7} + 9q^{8} + 2q^{9} + 7q^{10} - 4q^{11} + 5q^{12} - 10q^{14} - 3q^{15} + 3q^{16} + q^{17} + q^{18} + 6q^{19} + q^{20} - 3q^{21} - 2q^{22} + 4q^{23} + 9q^{24} + 3q^{25} + 2q^{27} - 16q^{28} + q^{29} + 7q^{30} + q^{31} + 9q^{32} - 4q^{33} + 9q^{34} - 4q^{35} + 5q^{36} + 11q^{37} - 14q^{38} - 5q^{40} + q^{41} - 10q^{42} + 5q^{43} - 10q^{44} - 3q^{45} + 2q^{46} + 3q^{48} - q^{49} - 24q^{50} + q^{51} + 11q^{53} + q^{54} + 6q^{55} - 22q^{56} + 6q^{57} - 25q^{58} - 14q^{59} + q^{60} + 16q^{61} - 8q^{62} - 3q^{63} + 7q^{64} - 2q^{66} + 5q^{67} + 11q^{68} + 4q^{69} - 2q^{70} + 28q^{71} + 9q^{72} - 12q^{73} - 3q^{74} + 3q^{75} - 2q^{76} + 6q^{77} + 15q^{79} + 21q^{80} + 2q^{81} + 9q^{82} - 10q^{83} - 16q^{84} + 7q^{85} - 6q^{86} + q^{87} - 18q^{88} + 18q^{89} + 7q^{90} + 10q^{92} + q^{93} - 34q^{94} - 26q^{95} + 9q^{96} - 13q^{97} + 25q^{98} - 4q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
−1.56155 1.00000 0.438447 −3.56155 −1.56155 0.561553 2.43845 1.00000 5.56155
1.2 2.56155 1.00000 4.56155 0.561553 2.56155 −3.56155 6.56155 1.00000 1.43845
 $$n$$: 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.g 2
3.b odd 2 1 1521.2.a.g 2
4.b odd 2 1 8112.2.a.bk 2
13.b even 2 1 507.2.a.d 2
13.c even 3 2 39.2.e.b 4
13.d odd 4 2 507.2.b.d 4
13.e even 6 2 507.2.e.g 4
13.f odd 12 4 507.2.j.g 8
39.d odd 2 1 1521.2.a.m 2
39.f even 4 2 1521.2.b.h 4
39.i odd 6 2 117.2.g.c 4
52.b odd 2 1 8112.2.a.bo 2
52.j odd 6 2 624.2.q.h 4
65.n even 6 2 975.2.i.k 4
65.q odd 12 4 975.2.bb.i 8
156.p even 6 2 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.c even 3 2
117.2.g.c 4 39.i odd 6 2
507.2.a.d 2 13.b even 2 1
507.2.a.g 2 1.a even 1 1 trivial
507.2.b.d 4 13.d odd 4 2
507.2.e.g 4 13.e even 6 2
507.2.j.g 8 13.f odd 12 4
624.2.q.h 4 52.j odd 6 2
975.2.i.k 4 65.n even 6 2
975.2.bb.i 8 65.q odd 12 4
1521.2.a.g 2 3.b odd 2 1
1521.2.a.m 2 39.d odd 2 1
1521.2.b.h 4 39.f even 4 2
1872.2.t.r 4 156.p even 6 2
8112.2.a.bk 2 4.b odd 2 1
8112.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 - T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-2 + 3 T + T^{2}$$
$7$ $$-2 + 3 T + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$-4 - T + T^{2}$$
$19$ $$-8 - 6 T + T^{2}$$
$23$ $$( -2 + T )^{2}$$
$29$ $$-38 - T + T^{2}$$
$31$ $$-4 - T + T^{2}$$
$37$ $$26 - 11 T + T^{2}$$
$41$ $$-4 - T + T^{2}$$
$43$ $$2 - 5 T + T^{2}$$
$47$ $$-68 + T^{2}$$
$53$ $$-8 - 11 T + T^{2}$$
$59$ $$32 + 14 T + T^{2}$$
$61$ $$47 - 16 T + T^{2}$$
$67$ $$2 - 5 T + T^{2}$$
$71$ $$( -14 + T )^{2}$$
$73$ $$19 + 12 T + T^{2}$$
$79$ $$52 - 15 T + T^{2}$$
$83$ $$8 + 10 T + T^{2}$$
$89$ $$64 - 18 T + T^{2}$$
$97$ $$38 + 13 T + T^{2}$$