# Properties

 Label 507.2.e.f Level $507$ Weight $2$ Character orbit 507.e Analytic conductor $4.048$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{2} q^{3} + ( 2 - 2 \zeta_{12}^{2} ) q^{4} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{5} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{2} q^{3} + ( 2 - 2 \zeta_{12}^{2} ) q^{4} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{5} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{11} + 2 q^{12} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} -4 \zeta_{12}^{2} q^{16} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{20} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{21} -6 \zeta_{12}^{2} q^{23} + 7 q^{25} - q^{27} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} -6 \zeta_{12}^{2} q^{29} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{31} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{33} + ( -6 + 6 \zeta_{12}^{2} ) q^{35} + 2 \zeta_{12}^{2} q^{36} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{41} + ( -1 + \zeta_{12}^{2} ) q^{43} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{44} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{45} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{47} + ( 4 - 4 \zeta_{12}^{2} ) q^{48} + 4 \zeta_{12}^{2} q^{49} + 12 q^{53} + 12 \zeta_{12}^{2} q^{55} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{59} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{60} + ( -1 + \zeta_{12}^{2} ) q^{61} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{63} -8 q^{64} + ( -5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{67} + ( 6 - 6 \zeta_{12}^{2} ) q^{69} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{71} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{73} + 7 \zeta_{12}^{2} q^{75} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{76} -6 q^{77} -11 q^{79} + ( 8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{80} -\zeta_{12}^{2} q^{81} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{83} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{84} + ( 6 - 6 \zeta_{12}^{2} ) q^{87} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} -12 q^{92} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{93} + ( -12 + 12 \zeta_{12}^{2} ) q^{95} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{97} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 4q^{4} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 4q^{4} - 2q^{9} + 8q^{12} - 8q^{16} - 12q^{23} + 28q^{25} - 4q^{27} - 12q^{29} - 12q^{35} + 4q^{36} - 2q^{43} + 8q^{48} + 8q^{49} + 48q^{53} + 24q^{55} - 2q^{61} - 32q^{64} + 12q^{69} + 14q^{75} - 24q^{77} - 44q^{79} - 2q^{81} + 12q^{87} - 48q^{92} - 24q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0.500000 0.866025i 1.00000 + 1.73205i −3.46410 0 0.866025 + 1.50000i 0 −0.500000 0.866025i 0
22.2 0 0.500000 0.866025i 1.00000 + 1.73205i 3.46410 0 −0.866025 1.50000i 0 −0.500000 0.866025i 0
484.1 0 0.500000 + 0.866025i 1.00000 1.73205i −3.46410 0 0.866025 1.50000i 0 −0.500000 + 0.866025i 0
484.2 0 0.500000 + 0.866025i 1.00000 1.73205i 3.46410 0 −0.866025 + 1.50000i 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.b even 2 1 inner
13.c even 3 1 inner
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.e.f 4
13.b even 2 1 inner 507.2.e.f 4
13.c even 3 1 507.2.a.e 2
13.c even 3 1 inner 507.2.e.f 4
13.d odd 4 1 39.2.j.a 2
13.d odd 4 1 507.2.j.b 2
13.e even 6 1 507.2.a.e 2
13.e even 6 1 inner 507.2.e.f 4
13.f odd 12 1 39.2.j.a 2
13.f odd 12 2 507.2.b.c 2
13.f odd 12 1 507.2.j.b 2
39.f even 4 1 117.2.q.a 2
39.h odd 6 1 1521.2.a.h 2
39.i odd 6 1 1521.2.a.h 2
39.k even 12 1 117.2.q.a 2
39.k even 12 2 1521.2.b.f 2
52.f even 4 1 624.2.bv.b 2
52.i odd 6 1 8112.2.a.bu 2
52.j odd 6 1 8112.2.a.bu 2
52.l even 12 1 624.2.bv.b 2
65.f even 4 1 975.2.w.d 4
65.g odd 4 1 975.2.bc.c 2
65.k even 4 1 975.2.w.d 4
65.o even 12 1 975.2.w.d 4
65.s odd 12 1 975.2.bc.c 2
65.t even 12 1 975.2.w.d 4
156.l odd 4 1 1872.2.by.f 2
156.v odd 12 1 1872.2.by.f 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( -12 + T^{2} )^{2}$$
$7$ $$9 + 3 T^{2} + T^{4}$$
$11$ $$144 + 12 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$144 + 12 T^{2} + T^{4}$$
$23$ $$( 36 + 6 T + T^{2} )^{2}$$
$29$ $$( 36 + 6 T + T^{2} )^{2}$$
$31$ $$( -3 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$2304 + 48 T^{2} + T^{4}$$
$43$ $$( 1 + T + T^{2} )^{2}$$
$47$ $$( -12 + T^{2} )^{2}$$
$53$ $$( -12 + T )^{4}$$
$59$ $$144 + 12 T^{2} + T^{4}$$
$61$ $$( 1 + T + T^{2} )^{2}$$
$67$ $$5625 + 75 T^{2} + T^{4}$$
$71$ $$11664 + 108 T^{2} + T^{4}$$
$73$ $$( -3 + T^{2} )^{2}$$
$79$ $$( 11 + T )^{4}$$
$83$ $$( -192 + T^{2} )^{2}$$
$89$ $$2304 + 48 T^{2} + T^{4}$$
$97$ $$729 + 27 T^{2} + T^{4}$$