# Properties

 Label 507.2.b.g Level $507$ Weight $2$ Character orbit 507.b Analytic conductor $4.048$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + \beta_{2} q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + \beta_{1} q^{6} + ( \beta_{3} - 3 \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + \beta_{2} q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + \beta_{1} q^{6} + ( \beta_{3} - 3 \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} + ( 1 - 2 \beta_{4} ) q^{10} + ( 3 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{11} + \beta_{2} q^{12} + ( -1 + 4 \beta_{4} ) q^{14} + ( -\beta_{3} + \beta_{5} ) q^{15} + ( -3 + 3 \beta_{2} + \beta_{4} ) q^{16} + ( 2 + \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( 3 \beta_{1} + \beta_{3} + 3 \beta_{5} ) q^{19} -\beta_{1} q^{20} + ( \beta_{3} - 3 \beta_{5} ) q^{21} + ( -2 + 3 \beta_{2} - 2 \beta_{4} ) q^{22} + ( 2 - 5 \beta_{2} - 3 \beta_{4} ) q^{23} + ( \beta_{1} + \beta_{3} ) q^{24} + ( 2 \beta_{2} + 3 \beta_{4} ) q^{25} + q^{27} + ( 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{28} + ( -1 - 2 \beta_{2} - 3 \beta_{4} ) q^{29} + ( 1 - 2 \beta_{4} ) q^{30} + ( -5 \beta_{1} + 2 \beta_{3} + 5 \beta_{5} ) q^{31} + ( -3 \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{32} + ( 3 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{33} + ( \beta_{1} + \beta_{3} ) q^{34} + ( 9 - 4 \beta_{2} - 5 \beta_{4} ) q^{35} + \beta_{2} q^{36} + ( -\beta_{1} + \beta_{3} - 4 \beta_{5} ) q^{37} + ( -7 + 3 \beta_{2} - 2 \beta_{4} ) q^{38} + ( 4 - \beta_{2} - 4 \beta_{4} ) q^{40} + \beta_{1} q^{41} + ( -1 + 4 \beta_{4} ) q^{42} + ( 1 - 2 \beta_{2} + 2 \beta_{4} ) q^{43} + ( -\beta_{1} - 3 \beta_{3} - 6 \beta_{5} ) q^{44} + ( -\beta_{3} + \beta_{5} ) q^{45} + ( 4 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} ) q^{46} + ( -9 \beta_{1} + 3 \beta_{3} + 7 \beta_{5} ) q^{47} + ( -3 + 3 \beta_{2} + \beta_{4} ) q^{48} + ( -10 + 6 \beta_{2} + 7 \beta_{4} ) q^{49} + ( \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{50} + ( 2 + \beta_{2} ) q^{51} + ( -6 + 2 \beta_{2} + 3 \beta_{4} ) q^{53} + \beta_{1} q^{54} + ( -5 + 2 \beta_{2} ) q^{55} + ( -6 + 3 \beta_{2} + 8 \beta_{4} ) q^{56} + ( 3 \beta_{1} + \beta_{3} + 3 \beta_{5} ) q^{57} + ( -2 \beta_{1} + \beta_{3} - 3 \beta_{5} ) q^{58} + ( -4 \beta_{1} + 6 \beta_{3} + 8 \beta_{5} ) q^{59} -\beta_{1} q^{60} + ( -4 - 3 \beta_{2} + 2 \beta_{4} ) q^{61} + ( 8 - 5 \beta_{2} - 3 \beta_{4} ) q^{62} + ( \beta_{3} - 3 \beta_{5} ) q^{63} + ( -4 + 3 \beta_{2} + 5 \beta_{4} ) q^{64} + ( -2 + 3 \beta_{2} - 2 \beta_{4} ) q^{66} + ( -4 \beta_{1} + 3 \beta_{3} + 4 \beta_{5} ) q^{67} + ( 1 + 3 \beta_{2} + \beta_{4} ) q^{68} + ( 2 - 5 \beta_{2} - 3 \beta_{4} ) q^{69} + ( 8 \beta_{1} + \beta_{3} - 5 \beta_{5} ) q^{70} + ( -5 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{71} + ( \beta_{1} + \beta_{3} ) q^{72} + ( 2 \beta_{1} - \beta_{3} - 7 \beta_{5} ) q^{73} + ( 1 - \beta_{2} + 5 \beta_{4} ) q^{74} + ( 2 \beta_{2} + 3 \beta_{4} ) q^{75} + ( -6 \beta_{1} + 7 \beta_{3} + 4 \beta_{5} ) q^{76} + ( 9 - 10 \beta_{2} - 2 \beta_{4} ) q^{77} + ( -5 + 5 \beta_{2} + \beta_{4} ) q^{79} + ( -\beta_{1} + 3 \beta_{3} - 4 \beta_{5} ) q^{80} + q^{81} + ( -2 + \beta_{2} ) q^{82} + ( -\beta_{1} - 3 \beta_{3} - 6 \beta_{5} ) q^{83} + ( 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{84} + ( -\beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{85} + ( 5 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} ) q^{86} + ( -1 - 2 \beta_{2} - 3 \beta_{4} ) q^{87} + ( 1 + 5 \beta_{2} - \beta_{4} ) q^{88} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{89} + ( 1 - 2 \beta_{4} ) q^{90} + ( -2 - 6 \beta_{2} - 5 \beta_{4} ) q^{92} + ( -5 \beta_{1} + 2 \beta_{3} + 5 \beta_{5} ) q^{93} + ( 15 - 9 \beta_{2} - 4 \beta_{4} ) q^{94} + ( 2 \beta_{2} - 5 \beta_{4} ) q^{95} + ( -3 \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{96} + ( 4 \beta_{1} - 10 \beta_{3} - 3 \beta_{5} ) q^{97} + ( -9 \beta_{1} - \beta_{3} + 7 \beta_{5} ) q^{98} + ( 3 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{3} + 2q^{4} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{3} + 2q^{4} + 6q^{9} + 2q^{10} + 2q^{12} + 2q^{14} - 10q^{16} + 14q^{17} - 10q^{22} - 4q^{23} + 10q^{25} + 6q^{27} - 16q^{29} + 2q^{30} + 36q^{35} + 2q^{36} - 40q^{38} + 14q^{40} + 2q^{42} + 6q^{43} - 10q^{48} - 34q^{49} + 14q^{51} - 26q^{53} - 26q^{55} - 14q^{56} - 26q^{61} + 32q^{62} - 8q^{64} - 10q^{66} + 14q^{68} - 4q^{69} + 14q^{74} + 10q^{75} + 30q^{77} - 18q^{79} + 6q^{81} - 10q^{82} - 16q^{87} + 14q^{88} + 2q^{90} - 34q^{92} + 64q^{94} - 6q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## 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
 − 1.80194i − 1.24698i − 0.445042i 0.445042i 1.24698i 1.80194i
1.80194i 1.00000 −1.24698 1.44504i 1.80194i 3.44504i 1.35690i 1.00000 −2.60388
337.2 1.24698i 1.00000 0.445042 2.80194i 1.24698i 4.80194i 3.04892i 1.00000 3.49396
337.3 0.445042i 1.00000 1.80194 0.246980i 0.445042i 1.75302i 1.69202i 1.00000 0.109916
337.4 0.445042i 1.00000 1.80194 0.246980i 0.445042i 1.75302i 1.69202i 1.00000 0.109916
337.5 1.24698i 1.00000 0.445042 2.80194i 1.24698i 4.80194i 3.04892i 1.00000 3.49396
337.6 1.80194i 1.00000 −1.24698 1.44504i 1.80194i 3.44504i 1.35690i 1.00000 −2.60388
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.6 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.g 6
3.b odd 2 1 1521.2.b.m 6
13.b even 2 1 inner 507.2.b.g 6
13.c even 3 2 507.2.j.h 12
13.d odd 4 1 507.2.a.j 3
13.d odd 4 1 507.2.a.k yes 3
13.e even 6 2 507.2.j.h 12
13.f odd 12 2 507.2.e.j 6
13.f odd 12 2 507.2.e.k 6
39.d odd 2 1 1521.2.b.m 6
39.f even 4 1 1521.2.a.p 3
39.f even 4 1 1521.2.a.q 3
52.f even 4 1 8112.2.a.by 3
52.f even 4 1 8112.2.a.cf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.d odd 4 1
507.2.a.k yes 3 13.d odd 4 1
507.2.b.g 6 1.a even 1 1 trivial
507.2.b.g 6 13.b even 2 1 inner
507.2.e.j 6 13.f odd 12 2
507.2.e.k 6 13.f odd 12 2
507.2.j.h 12 13.c even 3 2
507.2.j.h 12 13.e even 6 2
1521.2.a.p 3 39.f even 4 1
1521.2.a.q 3 39.f even 4 1
1521.2.b.m 6 3.b odd 2 1
1521.2.b.m 6 39.d odd 2 1
8112.2.a.by 3 52.f even 4 1
8112.2.a.cf 3 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{6} + 5 T_{2}^{4} + 6 T_{2}^{2} + 1$$ $$T_{5}^{6} + 10 T_{5}^{4} + 17 T_{5}^{2} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$3$ $$( -1 + T )^{6}$$
$5$ $$1 + 17 T^{2} + 10 T^{4} + T^{6}$$
$7$ $$841 + 381 T^{2} + 38 T^{4} + T^{6}$$
$11$ $$1849 + 986 T^{2} + 61 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( -7 + 14 T - 7 T^{2} + T^{3} )^{2}$$
$19$ $$12769 + 2586 T^{2} + 101 T^{4} + T^{6}$$
$23$ $$( 83 - 43 T + 2 T^{2} + T^{3} )^{2}$$
$29$ $$( -43 + 5 T + 8 T^{2} + T^{3} )^{2}$$
$31$ $$38809 + 3681 T^{2} + 110 T^{4} + T^{6}$$
$37$ $$8281 + 1421 T^{2} + 70 T^{4} + T^{6}$$
$41$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$43$ $$( -29 - 25 T - 3 T^{2} + T^{3} )^{2}$$
$47$ $$829921 + 30798 T^{2} + 321 T^{4} + T^{6}$$
$53$ $$( 29 + 40 T + 13 T^{2} + T^{3} )^{2}$$
$59$ $$3136 + 1568 T^{2} + 196 T^{4} + T^{6}$$
$61$ $$( -223 + 12 T + 13 T^{2} + T^{3} )^{2}$$
$67$ $$9409 + 1454 T^{2} + 69 T^{4} + T^{6}$$
$71$ $$212521 + 11773 T^{2} + 194 T^{4} + T^{6}$$
$73$ $$27889 + 4189 T^{2} + 122 T^{4} + T^{6}$$
$79$ $$( -169 - 22 T + 9 T^{2} + T^{3} )^{2}$$
$83$ $$1849 + 4401 T^{2} + 146 T^{4} + T^{6}$$
$89$ $$1 + 54 T^{2} + 41 T^{4} + T^{6}$$
$97$ $$1413721 + 40451 T^{2} + 363 T^{4} + T^{6}$$