# Properties

 Label 507.2.e.k Level $507$ Weight $2$ Character orbit 507.e 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.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{5} ) q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{6} + ( -\beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{3} ) q^{8} -\beta_{5} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{5} ) q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{6} + ( -\beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{3} ) q^{8} -\beta_{5} q^{9} + ( 1 - 2 \beta_{1} - \beta_{5} ) q^{10} + ( 2 - 3 \beta_{1} - 4 \beta_{4} - 2 \beta_{5} ) q^{11} + ( -1 + \beta_{2} - \beta_{3} ) q^{12} + ( -1 + 4 \beta_{2} ) q^{14} + ( 1 + \beta_{4} - \beta_{5} ) q^{15} + ( 2 \beta_{1} + 3 \beta_{4} ) q^{16} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{17} -\beta_{2} q^{18} + ( -3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{20} + ( -3 + \beta_{3} ) q^{21} + ( -5 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{22} + ( -3 + 2 \beta_{1} + 5 \beta_{4} + 3 \beta_{5} ) q^{23} + ( -\beta_{1} + \beta_{4} ) q^{24} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{25} + q^{27} + ( -2 + 3 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{28} + ( 3 + \beta_{1} - 2 \beta_{4} - 3 \beta_{5} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{30} + ( -5 + 5 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{32} + ( 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{33} + ( \beta_{2} + \beta_{3} ) q^{34} + ( -\beta_{1} + \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 5 \beta_{5} ) q^{35} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{36} + ( 4 + \beta_{1} + \beta_{4} - 4 \beta_{5} ) q^{37} + ( 4 + 5 \beta_{2} - 3 \beta_{3} ) q^{38} + ( 3 - 3 \beta_{2} - \beta_{3} ) q^{40} + \beta_{1} q^{41} + ( 1 - 4 \beta_{1} - \beta_{5} ) q^{42} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{43} + ( 6 + \beta_{2} + 3 \beta_{3} ) q^{44} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{45} + ( 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{46} + ( 7 - 9 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{48} + ( -4 + \beta_{1} - 6 \beta_{4} + 4 \beta_{5} ) q^{49} + ( -3 - \beta_{1} - \beta_{4} + 3 \beta_{5} ) q^{50} + ( -3 + \beta_{2} - \beta_{3} ) q^{51} + ( -4 + \beta_{2} + 2 \beta_{3} ) q^{53} + \beta_{1} q^{54} + ( 3 + 2 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} ) q^{55} + ( -5 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{56} + ( -3 - 3 \beta_{2} - \beta_{3} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{58} + ( -4 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 8 \beta_{5} ) q^{59} -\beta_{2} q^{60} + ( 5 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} ) q^{61} + ( 3 + 2 \beta_{1} + 5 \beta_{4} - 3 \beta_{5} ) q^{62} + ( 3 + \beta_{4} - 3 \beta_{5} ) q^{63} + ( 1 - 2 \beta_{2} - 3 \beta_{3} ) q^{64} + ( 1 - 5 \beta_{2} + 3 \beta_{3} ) q^{66} + ( 4 - 4 \beta_{1} - 3 \beta_{4} - 4 \beta_{5} ) q^{67} + ( -4 + 2 \beta_{1} + 3 \beta_{4} + 4 \beta_{5} ) q^{68} + ( -2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} ) q^{69} + ( 5 - 8 \beta_{2} - \beta_{3} ) q^{70} + ( 5 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{71} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{72} + ( -7 + 2 \beta_{2} - \beta_{3} ) q^{73} + ( 6 \beta_{1} - 6 \beta_{2} + \beta_{3} + \beta_{4} ) q^{74} + ( 2 + \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{75} + ( -4 + 6 \beta_{1} + 7 \beta_{4} + 4 \beta_{5} ) q^{76} + ( 1 - 8 \beta_{2} + 10 \beta_{3} ) q^{77} + ( -4 \beta_{2} + 5 \beta_{3} ) q^{79} + ( -4 - \beta_{1} - 3 \beta_{4} + 4 \beta_{5} ) q^{80} + ( -1 + \beta_{5} ) q^{81} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{82} + ( 6 + \beta_{2} + 3 \beta_{3} ) q^{83} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{84} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{85} + ( 2 + 5 \beta_{2} - 4 \beta_{3} ) q^{86} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{87} + ( 6 - 6 \beta_{1} - 5 \beta_{4} - 6 \beta_{5} ) q^{88} + ( -2 + 2 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{89} + ( -1 + 2 \beta_{2} ) q^{90} + ( -8 + \beta_{2} - 6 \beta_{3} ) q^{92} + ( 5 - 5 \beta_{1} - 2 \beta_{4} - 5 \beta_{5} ) q^{93} + ( -6 - 5 \beta_{1} - 9 \beta_{4} + 6 \beta_{5} ) q^{94} + ( 7 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{95} + ( -1 + 3 \beta_{2} - 4 \beta_{3} ) q^{96} + ( -4 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} - 3 \beta_{5} ) q^{97} + ( -9 \beta_{1} + 9 \beta_{2} + \beta_{3} + \beta_{4} - 7 \beta_{5} ) q^{98} + ( -2 + 3 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{2} - 3q^{3} + q^{4} - 8q^{5} + q^{6} + 10q^{7} - 3q^{9} + O(q^{10})$$ $$6q + q^{2} - 3q^{3} + q^{4} - 8q^{5} + q^{6} + 10q^{7} - 3q^{9} + q^{10} - q^{11} - 2q^{12} + 2q^{14} + 4q^{15} + 5q^{16} + 7q^{17} - 2q^{18} + 11q^{19} + q^{20} - 20q^{21} + 5q^{22} - 2q^{23} - 10q^{25} + 6q^{27} - q^{28} + 8q^{29} + q^{30} - 16q^{31} - 4q^{32} - q^{33} - 18q^{35} + q^{36} + 14q^{37} + 40q^{38} + 14q^{40} + q^{41} - q^{42} + 3q^{43} + 32q^{44} + 4q^{45} + 3q^{46} + 18q^{47} + 5q^{48} - 17q^{49} - 11q^{50} - 14q^{51} - 26q^{53} + q^{54} + 13q^{55} - 7q^{56} - 22q^{57} - 12q^{58} - 14q^{59} - 2q^{60} + 13q^{61} + 16q^{62} + 10q^{63} + 8q^{64} - 10q^{66} + 5q^{67} - 7q^{68} - 2q^{69} + 16q^{70} - 6q^{71} - 36q^{73} - 7q^{74} + 5q^{75} + q^{76} - 30q^{77} - 18q^{79} - 16q^{80} - 3q^{81} - 5q^{82} + 32q^{83} - q^{84} - 7q^{85} + 30q^{86} + 8q^{87} + 7q^{88} - 5q^{89} - 2q^{90} - 34q^{92} + 8q^{93} - 32q^{94} - 3q^{95} + 8q^{96} + 5q^{97} - 13q^{98} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}$$

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-\beta_{5}$$

## 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
 −0.623490 + 1.07992i 0.222521 − 0.385418i 0.900969 − 1.56052i −0.623490 − 1.07992i 0.222521 + 0.385418i 0.900969 + 1.56052i
−0.623490 + 1.07992i −0.500000 + 0.866025i 0.222521 + 0.385418i −2.80194 −0.623490 1.07992i 2.40097 + 4.15860i −3.04892 −0.500000 0.866025i 1.74698 3.02586i
22.2 0.222521 0.385418i −0.500000 + 0.866025i 0.900969 + 1.56052i 0.246980 0.222521 + 0.385418i 0.876510 + 1.51816i 1.69202 −0.500000 0.866025i 0.0549581 0.0951903i
22.3 0.900969 1.56052i −0.500000 + 0.866025i −0.623490 1.07992i −1.44504 0.900969 + 1.56052i 1.72252 + 2.98349i 1.35690 −0.500000 0.866025i −1.30194 + 2.25502i
484.1 −0.623490 1.07992i −0.500000 0.866025i 0.222521 0.385418i −2.80194 −0.623490 + 1.07992i 2.40097 4.15860i −3.04892 −0.500000 + 0.866025i 1.74698 + 3.02586i
484.2 0.222521 + 0.385418i −0.500000 0.866025i 0.900969 1.56052i 0.246980 0.222521 0.385418i 0.876510 1.51816i 1.69202 −0.500000 + 0.866025i 0.0549581 + 0.0951903i
484.3 0.900969 + 1.56052i −0.500000 0.866025i −0.623490 + 1.07992i −1.44504 0.900969 1.56052i 1.72252 2.98349i 1.35690 −0.500000 + 0.866025i −1.30194 2.25502i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 484.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.e.k 6
13.b even 2 1 507.2.e.j 6
13.c even 3 1 507.2.a.j 3
13.c even 3 1 inner 507.2.e.k 6
13.d odd 4 2 507.2.j.h 12
13.e even 6 1 507.2.a.k yes 3
13.e even 6 1 507.2.e.j 6
13.f odd 12 2 507.2.b.g 6
13.f odd 12 2 507.2.j.h 12
39.h odd 6 1 1521.2.a.p 3
39.i odd 6 1 1521.2.a.q 3
39.k even 12 2 1521.2.b.m 6
52.i odd 6 1 8112.2.a.cf 3
52.j odd 6 1 8112.2.a.by 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.c even 3 1
507.2.a.k yes 3 13.e even 6 1
507.2.b.g 6 13.f odd 12 2
507.2.e.j 6 13.b even 2 1
507.2.e.j 6 13.e even 6 1
507.2.e.k 6 1.a even 1 1 trivial
507.2.e.k 6 13.c even 3 1 inner
507.2.j.h 12 13.d odd 4 2
507.2.j.h 12 13.f odd 12 2
1521.2.a.p 3 39.h odd 6 1
1521.2.a.q 3 39.i odd 6 1
1521.2.b.m 6 39.k even 12 2
8112.2.a.by 3 52.j odd 6 1
8112.2.a.cf 3 52.i odd 6 1

## Hecke kernels

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

 $$T_{2}^{6} - T_{2}^{5} + 3 T_{2}^{4} + 5 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{5}^{3} + 4 T_{5}^{2} + 3 T_{5} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6}$$
$3$ $$( 1 + T + T^{2} )^{3}$$
$5$ $$( -1 + 3 T + 4 T^{2} + T^{3} )^{2}$$
$7$ $$841 - 899 T + 671 T^{2} - 252 T^{3} + 69 T^{4} - 10 T^{5} + T^{6}$$
$11$ $$1849 + 1290 T + 943 T^{2} + 56 T^{3} + 31 T^{4} + T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$49 - 98 T + 147 T^{2} - 84 T^{3} + 35 T^{4} - 7 T^{5} + T^{6}$$
$19$ $$12769 + 1130 T + 1343 T^{2} - 336 T^{3} + 111 T^{4} - 11 T^{5} + T^{6}$$
$23$ $$6889 - 3569 T + 1683 T^{2} - 252 T^{3} + 47 T^{4} + 2 T^{5} + T^{6}$$
$29$ $$1849 + 215 T + 369 T^{2} - 126 T^{3} + 59 T^{4} - 8 T^{5} + T^{6}$$
$31$ $$( -197 - 23 T + 8 T^{2} + T^{3} )^{2}$$
$37$ $$8281 - 5733 T + 2695 T^{2} - 700 T^{3} + 133 T^{4} - 14 T^{5} + T^{6}$$
$41$ $$1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6}$$
$43$ $$841 + 725 T + 538 T^{2} + 133 T^{3} + 34 T^{4} - 3 T^{5} + T^{6}$$
$47$ $$( 911 - 120 T - 9 T^{2} + T^{3} )^{2}$$
$53$ $$( 29 + 40 T + 13 T^{2} + T^{3} )^{2}$$
$59$ $$3136 + 784 T^{2} + 112 T^{3} + 196 T^{4} + 14 T^{5} + T^{6}$$
$61$ $$49729 + 2676 T + 3043 T^{2} - 602 T^{3} + 157 T^{4} - 13 T^{5} + T^{6}$$
$67$ $$9409 - 2134 T + 969 T^{2} - 84 T^{3} + 47 T^{4} - 5 T^{5} + T^{6}$$
$71$ $$212521 + 36419 T + 9007 T^{2} + 448 T^{3} + 115 T^{4} + 6 T^{5} + T^{6}$$
$73$ $$( 167 + 101 T + 18 T^{2} + T^{3} )^{2}$$
$79$ $$( -169 - 22 T + 9 T^{2} + T^{3} )^{2}$$
$83$ $$( 43 + 55 T - 16 T^{2} + T^{3} )^{2}$$
$89$ $$1 - 8 T + 59 T^{2} - 42 T^{3} + 33 T^{4} + 5 T^{5} + T^{6}$$
$97$ $$1413721 - 200941 T + 34506 T^{2} - 1533 T^{3} + 194 T^{4} - 5 T^{5} + T^{6}$$