# Properties

 Label 507.2.a.e Level $507$ Weight $2$ Character orbit 507.a Self dual yes Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} -2 q^{4} + 2 \beta q^{5} + \beta q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -2 q^{4} + 2 \beta q^{5} + \beta q^{7} + q^{9} -2 \beta q^{11} + 2 q^{12} -2 \beta q^{15} + 4 q^{16} + 2 \beta q^{19} -4 \beta q^{20} -\beta q^{21} + 6 q^{23} + 7 q^{25} - q^{27} -2 \beta q^{28} + 6 q^{29} + \beta q^{31} + 2 \beta q^{33} + 6 q^{35} -2 q^{36} + 4 \beta q^{41} + q^{43} + 4 \beta q^{44} + 2 \beta q^{45} -2 \beta q^{47} -4 q^{48} -4 q^{49} + 12 q^{53} -12 q^{55} -2 \beta q^{57} + 2 \beta q^{59} + 4 \beta q^{60} + q^{61} + \beta q^{63} -8 q^{64} -5 \beta q^{67} -6 q^{69} -6 \beta q^{71} -\beta q^{73} -7 q^{75} -4 \beta q^{76} -6 q^{77} -11 q^{79} + 8 \beta q^{80} + q^{81} -8 \beta q^{83} + 2 \beta q^{84} -6 q^{87} + 4 \beta q^{89} -12 q^{92} -\beta q^{93} + 12 q^{95} -3 \beta q^{97} -2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 4q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 4q^{4} + 2q^{9} + 4q^{12} + 8q^{16} + 12q^{23} + 14q^{25} - 2q^{27} + 12q^{29} + 12q^{35} - 4q^{36} + 2q^{43} - 8q^{48} - 8q^{49} + 24q^{53} - 24q^{55} + 2q^{61} - 16q^{64} - 12q^{69} - 14q^{75} - 12q^{77} - 22q^{79} + 2q^{81} - 12q^{87} - 24q^{92} + 24q^{95} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −1.00000 −2.00000 −3.46410 0 −1.73205 0 1.00000 0
1.2 0 −1.00000 −2.00000 3.46410 0 1.73205 0 1.00000 0
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.a.e 2
3.b odd 2 1 1521.2.a.h 2
4.b odd 2 1 8112.2.a.bu 2
13.b even 2 1 inner 507.2.a.e 2
13.c even 3 2 507.2.e.f 4
13.d odd 4 2 507.2.b.c 2
13.e even 6 2 507.2.e.f 4
13.f odd 12 2 39.2.j.a 2
13.f odd 12 2 507.2.j.b 2
39.d odd 2 1 1521.2.a.h 2
39.f even 4 2 1521.2.b.f 2
39.k even 12 2 117.2.q.a 2
52.b odd 2 1 8112.2.a.bu 2
52.l even 12 2 624.2.bv.b 2
65.o even 12 2 975.2.w.d 4
65.s odd 12 2 975.2.bc.c 2
65.t even 12 2 975.2.w.d 4
156.v odd 12 2 1872.2.by.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 13.f odd 12 2
117.2.q.a 2 39.k even 12 2
507.2.a.e 2 1.a even 1 1 trivial
507.2.a.e 2 13.b even 2 1 inner
507.2.b.c 2 13.d odd 4 2
507.2.e.f 4 13.c even 3 2
507.2.e.f 4 13.e even 6 2
507.2.j.b 2 13.f odd 12 2
624.2.bv.b 2 52.l even 12 2
975.2.w.d 4 65.o even 12 2
975.2.w.d 4 65.t even 12 2
975.2.bc.c 2 65.s odd 12 2
1521.2.a.h 2 3.b odd 2 1
1521.2.a.h 2 39.d odd 2 1
1521.2.b.f 2 39.f even 4 2
1872.2.by.f 2 156.v odd 12 2
8112.2.a.bu 2 4.b odd 2 1
8112.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

 $$T_{2}$$ $$T_{5}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-12 + T^{2}$$
$7$ $$-3 + T^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$-12 + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$-3 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$-48 + T^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$-12 + T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$-12 + T^{2}$$
$61$ $$( -1 + T )^{2}$$
$67$ $$-75 + T^{2}$$
$71$ $$-108 + T^{2}$$
$73$ $$-3 + T^{2}$$
$79$ $$( 11 + T )^{2}$$
$83$ $$-192 + T^{2}$$
$89$ $$-48 + T^{2}$$
$97$ $$-27 + T^{2}$$