# Properties

 Label 507.2.j.b Level $507$ Weight $2$ Character orbit 507.j Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 + 2 \zeta_{6} ) q^{11} -2 q^{12} + ( -2 - 2 \zeta_{6} ) q^{15} + ( -4 + 4 \zeta_{6} ) q^{16} + ( -4 + 2 \zeta_{6} ) q^{19} + ( -8 + 4 \zeta_{6} ) q^{20} + ( 1 - 2 \zeta_{6} ) q^{21} + ( 6 - 6 \zeta_{6} ) q^{23} -7 q^{25} - q^{27} + ( -2 - 2 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + ( 1 - 2 \zeta_{6} ) q^{31} + ( 4 - 2 \zeta_{6} ) q^{33} -6 \zeta_{6} q^{35} + ( -2 + 2 \zeta_{6} ) q^{36} + ( 4 + 4 \zeta_{6} ) q^{41} + \zeta_{6} q^{43} + ( 4 - 8 \zeta_{6} ) q^{44} + ( -4 + 2 \zeta_{6} ) q^{45} + ( 2 - 4 \zeta_{6} ) q^{47} + 4 \zeta_{6} q^{48} + ( -4 + 4 \zeta_{6} ) q^{49} + 12 q^{53} + ( 12 - 12 \zeta_{6} ) q^{55} + ( -2 + 4 \zeta_{6} ) q^{57} + ( 4 - 2 \zeta_{6} ) q^{59} + ( -4 + 8 \zeta_{6} ) q^{60} -\zeta_{6} q^{61} + ( -1 - \zeta_{6} ) q^{63} + 8 q^{64} + ( -5 - 5 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{69} + ( 12 - 6 \zeta_{6} ) q^{71} + ( 1 - 2 \zeta_{6} ) q^{73} + ( -7 + 7 \zeta_{6} ) q^{75} + ( 4 + 4 \zeta_{6} ) q^{76} + 6 q^{77} -11 q^{79} + ( 8 + 8 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + ( -8 + 16 \zeta_{6} ) q^{83} + ( -4 + 2 \zeta_{6} ) q^{84} + 6 \zeta_{6} q^{87} + ( -4 - 4 \zeta_{6} ) q^{89} -12 q^{92} + ( -1 - \zeta_{6} ) q^{93} + 12 \zeta_{6} q^{95} + ( 6 - 3 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 2q^{4} + 3q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 2q^{4} + 3q^{7} - q^{9} + 6q^{11} - 4q^{12} - 6q^{15} - 4q^{16} - 6q^{19} - 12q^{20} + 6q^{23} - 14q^{25} - 2q^{27} - 6q^{28} - 6q^{29} + 6q^{33} - 6q^{35} - 2q^{36} + 12q^{41} + q^{43} - 6q^{45} + 4q^{48} - 4q^{49} + 24q^{53} + 12q^{55} + 6q^{59} - q^{61} - 3q^{63} + 16q^{64} - 15q^{67} - 6q^{69} + 18q^{71} - 7q^{75} + 12q^{76} + 12q^{77} - 22q^{79} + 24q^{80} - q^{81} - 6q^{84} + 6q^{87} - 12q^{89} - 24q^{92} - 3q^{93} + 12q^{95} + 9q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i −1.00000 1.73205i 3.46410i 0 1.50000 0.866025i 0 −0.500000 0.866025i 0
361.1 0 0.500000 + 0.866025i −1.00000 + 1.73205i 3.46410i 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

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

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$12 + T^{2}$$
$7$ $$3 - 3 T + T^{2}$$
$11$ $$12 - 6 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$12 + 6 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$3 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$48 - 12 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$12 + T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$12 - 6 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$75 + 15 T + T^{2}$$
$71$ $$108 - 18 T + T^{2}$$
$73$ $$3 + T^{2}$$
$79$ $$( 11 + T )^{2}$$
$83$ $$192 + T^{2}$$
$89$ $$48 + 12 T + T^{2}$$
$97$ $$27 - 9 T + T^{2}$$
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