# Properties

 Label 507.2.b.d Level $507$ Weight $2$ Character orbit 507.b Analytic conductor $4.048$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + ( -3 + \beta_{3} ) q^{4} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + ( -3 + \beta_{3} ) q^{4} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} + q^{9} + ( -3 - \beta_{3} ) q^{10} -2 \beta_{2} q^{11} + ( -3 + \beta_{3} ) q^{12} + ( -6 + 2 \beta_{3} ) q^{14} + ( \beta_{1} + 2 \beta_{2} ) q^{15} + ( 3 - 3 \beta_{3} ) q^{16} + ( -1 + \beta_{3} ) q^{17} + \beta_{1} q^{18} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{19} -\beta_{1} q^{20} + ( \beta_{1} - \beta_{2} ) q^{21} + ( -2 + 2 \beta_{3} ) q^{22} -2 q^{23} + ( -\beta_{1} + 4 \beta_{2} ) q^{24} -3 \beta_{3} q^{25} + q^{27} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{28} + ( -1 + 3 \beta_{3} ) q^{29} + ( -3 - \beta_{3} ) q^{30} + ( -\beta_{1} - \beta_{2} ) q^{31} + ( \beta_{1} - 4 \beta_{2} ) q^{32} -2 \beta_{2} q^{33} + ( -\beta_{1} + 4 \beta_{2} ) q^{34} -2 q^{35} + ( -3 + \beta_{3} ) q^{36} + ( \beta_{1} + 6 \beta_{2} ) q^{37} + ( 6 + 2 \beta_{3} ) q^{38} + ( -1 - 3 \beta_{3} ) q^{40} + \beta_{1} q^{41} + ( -6 + 2 \beta_{3} ) q^{42} + ( -2 - \beta_{3} ) q^{43} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{44} + ( \beta_{1} + 2 \beta_{2} ) q^{45} -2 \beta_{1} q^{46} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 3 - 3 \beta_{3} ) q^{48} + ( -1 + 3 \beta_{3} ) q^{49} -12 \beta_{2} q^{50} + ( -1 + \beta_{3} ) q^{51} + ( 7 - 3 \beta_{3} ) q^{53} + \beta_{1} q^{54} + ( 2 + 2 \beta_{3} ) q^{55} + ( 14 - 6 \beta_{3} ) q^{56} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{57} + ( -\beta_{1} + 12 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{59} -\beta_{1} q^{60} + ( 9 - 2 \beta_{3} ) q^{61} + 4 q^{62} + ( \beta_{1} - \beta_{2} ) q^{63} + ( -3 - \beta_{3} ) q^{64} + ( -2 + 2 \beta_{3} ) q^{66} + ( -\beta_{1} - 3 \beta_{2} ) q^{67} + ( 7 - 3 \beta_{3} ) q^{68} -2 q^{69} -2 \beta_{1} q^{70} -14 \beta_{2} q^{71} + ( -\beta_{1} + 4 \beta_{2} ) q^{72} + ( -2 \beta_{1} - 7 \beta_{2} ) q^{73} + ( 1 - 5 \beta_{3} ) q^{74} -3 \beta_{3} q^{75} + 2 \beta_{1} q^{76} + ( -4 + 2 \beta_{3} ) q^{77} + ( 8 - \beta_{3} ) q^{79} + ( -3 \beta_{1} - 12 \beta_{2} ) q^{80} + q^{81} + ( -5 + \beta_{3} ) q^{82} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{84} + ( \beta_{1} + 4 \beta_{2} ) q^{85} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{86} + ( -1 + 3 \beta_{3} ) q^{87} + ( 10 - 2 \beta_{3} ) q^{88} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{89} + ( -3 - \beta_{3} ) q^{90} + ( 6 - 2 \beta_{3} ) q^{92} + ( -\beta_{1} - \beta_{2} ) q^{93} + ( -18 + 2 \beta_{3} ) q^{94} + ( 10 + 6 \beta_{3} ) q^{95} + ( \beta_{1} - 4 \beta_{2} ) q^{96} + ( \beta_{1} + 7 \beta_{2} ) q^{97} + ( -\beta_{1} + 12 \beta_{2} ) q^{98} -2 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 10q^{4} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 10q^{4} + 4q^{9} - 14q^{10} - 10q^{12} - 20q^{14} + 6q^{16} - 2q^{17} - 4q^{22} - 8q^{23} - 6q^{25} + 4q^{27} + 2q^{29} - 14q^{30} - 8q^{35} - 10q^{36} + 28q^{38} - 10q^{40} - 20q^{42} - 10q^{43} + 6q^{48} + 2q^{49} - 2q^{51} + 22q^{53} + 12q^{55} + 44q^{56} + 32q^{61} + 16q^{62} - 14q^{64} - 4q^{66} + 22q^{68} - 8q^{69} - 6q^{74} - 6q^{75} - 12q^{77} + 30q^{79} + 4q^{81} - 18q^{82} + 2q^{87} + 36q^{88} - 14q^{90} + 20q^{92} - 68q^{94} + 52q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

## 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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 1.00000 −4.56155 0.561553i 2.56155i 3.56155i 6.56155i 1.00000 −1.43845
337.2 1.56155i 1.00000 −0.438447 3.56155i 1.56155i 0.561553i 2.43845i 1.00000 −5.56155
337.3 1.56155i 1.00000 −0.438447 3.56155i 1.56155i 0.561553i 2.43845i 1.00000 −5.56155
337.4 2.56155i 1.00000 −4.56155 0.561553i 2.56155i 3.56155i 6.56155i 1.00000 −1.43845
 $$n$$: 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.d 4
3.b odd 2 1 1521.2.b.h 4
13.b even 2 1 inner 507.2.b.d 4
13.c even 3 2 507.2.j.g 8
13.d odd 4 1 507.2.a.d 2
13.d odd 4 1 507.2.a.g 2
13.e even 6 2 507.2.j.g 8
13.f odd 12 2 39.2.e.b 4
13.f odd 12 2 507.2.e.g 4
39.d odd 2 1 1521.2.b.h 4
39.f even 4 1 1521.2.a.g 2
39.f even 4 1 1521.2.a.m 2
39.k even 12 2 117.2.g.c 4
52.f even 4 1 8112.2.a.bk 2
52.f even 4 1 8112.2.a.bo 2
52.l even 12 2 624.2.q.h 4
65.o even 12 2 975.2.bb.i 8
65.s odd 12 2 975.2.i.k 4
65.t even 12 2 975.2.bb.i 8
156.v odd 12 2 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.f odd 12 2
117.2.g.c 4 39.k even 12 2
507.2.a.d 2 13.d odd 4 1
507.2.a.g 2 13.d odd 4 1
507.2.b.d 4 1.a even 1 1 trivial
507.2.b.d 4 13.b even 2 1 inner
507.2.e.g 4 13.f odd 12 2
507.2.j.g 8 13.c even 3 2
507.2.j.g 8 13.e even 6 2
624.2.q.h 4 52.l even 12 2
975.2.i.k 4 65.s odd 12 2
975.2.bb.i 8 65.o even 12 2
975.2.bb.i 8 65.t even 12 2
1521.2.a.g 2 39.f even 4 1
1521.2.a.m 2 39.f even 4 1
1521.2.b.h 4 3.b odd 2 1
1521.2.b.h 4 39.d odd 2 1
1872.2.t.r 4 156.v odd 12 2
8112.2.a.bk 2 52.f even 4 1
8112.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{4} + 9 T_{2}^{2} + 16$$ $$T_{5}^{4} + 13 T_{5}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 9 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$4 + 13 T^{2} + T^{4}$$
$7$ $$4 + 13 T^{2} + T^{4}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -4 + T + T^{2} )^{2}$$
$19$ $$64 + 52 T^{2} + T^{4}$$
$23$ $$( 2 + T )^{4}$$
$29$ $$( -38 - T + T^{2} )^{2}$$
$31$ $$16 + 9 T^{2} + T^{4}$$
$37$ $$676 + 69 T^{2} + T^{4}$$
$41$ $$16 + 9 T^{2} + T^{4}$$
$43$ $$( 2 + 5 T + T^{2} )^{2}$$
$47$ $$( 68 + T^{2} )^{2}$$
$53$ $$( -8 - 11 T + T^{2} )^{2}$$
$59$ $$1024 + 132 T^{2} + T^{4}$$
$61$ $$( 47 - 16 T + T^{2} )^{2}$$
$67$ $$4 + 21 T^{2} + T^{4}$$
$71$ $$( 196 + T^{2} )^{2}$$
$73$ $$361 + 106 T^{2} + T^{4}$$
$79$ $$( 52 - 15 T + T^{2} )^{2}$$
$83$ $$64 + 84 T^{2} + T^{4}$$
$89$ $$4096 + 196 T^{2} + T^{4}$$
$97$ $$1444 + 93 T^{2} + T^{4}$$