Properties

 Label 507.2.j.c Level $507$ Weight $2$ Character orbit 507.j 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.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: $$\Z[a_1, a_2]$$ 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.

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

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
1.50000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 0 1.50000 0.866025i 3.00000 1.73205i 1.73205i −0.500000 0.866025i 0
361.1 1.50000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 0 1.50000 + 0.866025i 3.00000 + 1.73205i 1.73205i −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.c 2
13.b even 2 1 507.2.j.a 2
13.c even 3 1 39.2.b.a 2
13.c even 3 1 507.2.j.a 2
13.d odd 4 2 507.2.e.e 4
13.e even 6 1 39.2.b.a 2
13.e even 6 1 inner 507.2.j.c 2
13.f odd 12 2 507.2.a.f 2
13.f odd 12 2 507.2.e.e 4
39.h odd 6 1 117.2.b.a 2
39.i odd 6 1 117.2.b.a 2
39.k even 12 2 1521.2.a.l 2
52.i odd 6 1 624.2.c.e 2
52.j odd 6 1 624.2.c.e 2
52.l even 12 2 8112.2.a.bv 2
65.l even 6 1 975.2.b.d 2
65.n even 6 1 975.2.b.d 2
65.q odd 12 2 975.2.h.f 4
65.r odd 12 2 975.2.h.f 4
91.n odd 6 1 1911.2.c.d 2
91.t odd 6 1 1911.2.c.d 2
104.n odd 6 1 2496.2.c.d 2
104.p odd 6 1 2496.2.c.d 2
104.r even 6 1 2496.2.c.k 2
104.s even 6 1 2496.2.c.k 2
156.p even 6 1 1872.2.c.e 2
156.r even 6 1 1872.2.c.e 2

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 - 3 T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$12 - 6 T + T^{2}$$
$11$ $$12 + 6 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$12 + 6 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$12 + T^{2}$$
$37$ $$48 - 12 T + T^{2}$$
$41$ $$48 + 12 T + T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$12 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$108 + 18 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$108 + 18 T + T^{2}$$
$71$ $$12 + 6 T + T^{2}$$
$73$ $$T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$48 + 12 T + T^{2}$$
$97$ $$192 - 24 T + T^{2}$$