# Properties

 Label 507.2.a.f 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: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
−1.73205 −1.00000 1.00000 0 1.73205 −3.46410 1.73205 1.00000 0
1.2 1.73205 −1.00000 1.00000 0 −1.73205 3.46410 −1.73205 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.f 2
3.b odd 2 1 1521.2.a.l 2
4.b odd 2 1 8112.2.a.bv 2
13.b even 2 1 inner 507.2.a.f 2
13.c even 3 2 507.2.e.e 4
13.d odd 4 2 39.2.b.a 2
13.e even 6 2 507.2.e.e 4
13.f odd 12 2 507.2.j.a 2
13.f odd 12 2 507.2.j.c 2
39.d odd 2 1 1521.2.a.l 2
39.f even 4 2 117.2.b.a 2
52.b odd 2 1 8112.2.a.bv 2
52.f even 4 2 624.2.c.e 2
65.f even 4 2 975.2.h.f 4
65.g odd 4 2 975.2.b.d 2
65.k even 4 2 975.2.h.f 4
91.i even 4 2 1911.2.c.d 2
104.j odd 4 2 2496.2.c.k 2
104.m even 4 2 2496.2.c.d 2
156.l odd 4 2 1872.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 13.d odd 4 2
117.2.b.a 2 39.f even 4 2
507.2.a.f 2 1.a even 1 1 trivial
507.2.a.f 2 13.b even 2 1 inner
507.2.e.e 4 13.c even 3 2
507.2.e.e 4 13.e even 6 2
507.2.j.a 2 13.f odd 12 2
507.2.j.c 2 13.f odd 12 2
624.2.c.e 2 52.f even 4 2
975.2.b.d 2 65.g odd 4 2
975.2.h.f 4 65.f even 4 2
975.2.h.f 4 65.k even 4 2
1521.2.a.l 2 3.b odd 2 1
1521.2.a.l 2 39.d odd 2 1
1872.2.c.e 2 156.l odd 4 2
1911.2.c.d 2 91.i even 4 2
2496.2.c.d 2 104.m even 4 2
2496.2.c.k 2 104.j odd 4 2
8112.2.a.bv 2 4.b odd 2 1
8112.2.a.bv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-12 + T^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$-12 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$-12 + T^{2}$$
$37$ $$-48 + T^{2}$$
$41$ $$-48 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$-12 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-108 + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$-108 + T^{2}$$
$71$ $$-12 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$-12 + T^{2}$$
$89$ $$-48 + T^{2}$$
$97$ $$-192 + T^{2}$$