Properties

 Label 507.2.b.b Level $507$ Weight $2$ Character orbit 507.b 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.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

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

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{3} + q^{4} -i q^{5} -i q^{6} -2 i q^{7} + 3 i q^{8} + q^{9} +O(q^{10})$$ $$q + i q^{2} - q^{3} + q^{4} -i q^{5} -i q^{6} -2 i q^{7} + 3 i q^{8} + q^{9} + q^{10} + 2 i q^{11} - q^{12} + 2 q^{14} + i q^{15} - q^{16} + 7 q^{17} + i q^{18} -6 i q^{19} -i q^{20} + 2 i q^{21} -2 q^{22} + 6 q^{23} -3 i q^{24} + 4 q^{25} - q^{27} -2 i q^{28} - q^{29} - q^{30} + 4 i q^{31} + 5 i q^{32} -2 i q^{33} + 7 i q^{34} -2 q^{35} + q^{36} -i q^{37} + 6 q^{38} + 3 q^{40} + 9 i q^{41} -2 q^{42} -6 q^{43} + 2 i q^{44} -i q^{45} + 6 i q^{46} -6 i q^{47} + q^{48} + 3 q^{49} + 4 i q^{50} -7 q^{51} -9 q^{53} -i q^{54} + 2 q^{55} + 6 q^{56} + 6 i q^{57} -i q^{58} + i q^{60} + q^{61} -4 q^{62} -2 i q^{63} -7 q^{64} + 2 q^{66} -2 i q^{67} + 7 q^{68} -6 q^{69} -2 i q^{70} + 6 i q^{71} + 3 i q^{72} -11 i q^{73} + q^{74} -4 q^{75} -6 i q^{76} + 4 q^{77} -4 q^{79} + i q^{80} + q^{81} -9 q^{82} -14 i q^{83} + 2 i q^{84} -7 i q^{85} -6 i q^{86} + q^{87} -6 q^{88} + 14 i q^{89} + q^{90} + 6 q^{92} -4 i q^{93} + 6 q^{94} -6 q^{95} -5 i q^{96} -2 i q^{97} + 3 i q^{98} + 2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{4} + 2q^{9} + 2q^{10} - 2q^{12} + 4q^{14} - 2q^{16} + 14q^{17} - 4q^{22} + 12q^{23} + 8q^{25} - 2q^{27} - 2q^{29} - 2q^{30} - 4q^{35} + 2q^{36} + 12q^{38} + 6q^{40} - 4q^{42} - 12q^{43} + 2q^{48} + 6q^{49} - 14q^{51} - 18q^{53} + 4q^{55} + 12q^{56} + 2q^{61} - 8q^{62} - 14q^{64} + 4q^{66} + 14q^{68} - 12q^{69} + 2q^{74} - 8q^{75} + 8q^{77} - 8q^{79} + 2q^{81} - 18q^{82} + 2q^{87} - 12q^{88} + 2q^{90} + 12q^{92} + 12q^{94} - 12q^{95} + O(q^{100})$$

Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
1.00000i −1.00000 1.00000 1.00000i 1.00000i 2.00000i 3.00000i 1.00000 1.00000
337.2 1.00000i −1.00000 1.00000 1.00000i 1.00000i 2.00000i 3.00000i 1.00000 1.00000
 $$n$$: 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

Twists

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

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

Hecke kernels

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

 $$T_{2}^{2} + 1$$ $$T_{5}^{2} + 1$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -7 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$81 + T^{2}$$
$43$ $$( 6 + T )^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$( 9 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$( -1 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$36 + T^{2}$$
$73$ $$121 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$196 + T^{2}$$
$89$ $$196 + T^{2}$$
$97$ $$4 + T^{2}$$