# Properties

 Label 507.2.e.i Level $507$ Weight $2$ Character orbit 507.e Analytic conductor $4.048$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Learn more

## 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.

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

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
−1.34601 + 2.33136i 0.500000 0.866025i −2.62349 4.54402i −1.04892 1.34601 + 2.33136i −0.277479 0.480608i 8.74094 −0.500000 0.866025i 1.41185 2.44540i
22.2 −1.17845 + 2.04113i 0.500000 0.866025i −1.77748 3.07868i 3.69202 1.17845 + 2.04113i 0.400969 + 0.694498i 3.66487 −0.500000 0.866025i −4.35086 + 7.53590i
22.3 1.02446 1.77441i 0.500000 0.866025i −1.09903 1.90358i 3.35690 −1.02446 1.77441i −1.12349 1.94594i −0.405813 −0.500000 0.866025i 3.43900 5.95652i
484.1 −1.34601 2.33136i 0.500000 + 0.866025i −2.62349 + 4.54402i −1.04892 1.34601 2.33136i −0.277479 + 0.480608i 8.74094 −0.500000 + 0.866025i 1.41185 + 2.44540i
484.2 −1.17845 2.04113i 0.500000 + 0.866025i −1.77748 + 3.07868i 3.69202 1.17845 2.04113i 0.400969 0.694498i 3.66487 −0.500000 + 0.866025i −4.35086 7.53590i
484.3 1.02446 + 1.77441i 0.500000 + 0.866025i −1.09903 + 1.90358i 3.35690 −1.02446 + 1.77441i −1.12349 + 1.94594i −0.405813 −0.500000 + 0.866025i 3.43900 + 5.95652i
 $$n$$: 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.i 6
13.b even 2 1 507.2.e.l 6
13.c even 3 1 507.2.a.l yes 3
13.c even 3 1 inner 507.2.e.i 6
13.d odd 4 2 507.2.j.i 12
13.e even 6 1 507.2.a.i 3
13.e even 6 1 507.2.e.l 6
13.f odd 12 2 507.2.b.f 6
13.f odd 12 2 507.2.j.i 12
39.h odd 6 1 1521.2.a.s 3
39.i odd 6 1 1521.2.a.n 3
39.k even 12 2 1521.2.b.k 6
52.i odd 6 1 8112.2.a.cg 3
52.j odd 6 1 8112.2.a.cp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 13.e even 6 1
507.2.a.l yes 3 13.c even 3 1
507.2.b.f 6 13.f odd 12 2
507.2.e.i 6 1.a even 1 1 trivial
507.2.e.i 6 13.c even 3 1 inner
507.2.e.l 6 13.b even 2 1
507.2.e.l 6 13.e even 6 1
507.2.j.i 12 13.d odd 4 2
507.2.j.i 12 13.f odd 12 2
1521.2.a.n 3 39.i odd 6 1
1521.2.a.s 3 39.h odd 6 1
1521.2.b.k 6 39.k even 12 2
8112.2.a.cg 3 52.i odd 6 1
8112.2.a.cp 3 52.j odd 6 1

## Hecke kernels

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

 $$T_{2}^{6} + 3 T_{2}^{5} + 13 T_{2}^{4} + 14 T_{2}^{3} + 55 T_{2}^{2} + 52 T_{2} + 169$$ $$T_{5}^{3} - 6 T_{5}^{2} + 5 T_{5} + 13$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$169 + 52 T + 55 T^{2} + 14 T^{3} + 13 T^{4} + 3 T^{5} + T^{6}$$
$3$ $$( 1 - T + T^{2} )^{3}$$
$5$ $$( 13 + 5 T - 6 T^{2} + T^{3} )^{2}$$
$7$ $$1 + T + 3 T^{2} + 5 T^{4} + 2 T^{5} + T^{6}$$
$11$ $$1681 + 328 T + 269 T^{2} + 42 T^{3} + 33 T^{4} + 5 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$169 + 208 T + 243 T^{2} + 42 T^{3} + 17 T^{4} - T^{5} + T^{6}$$
$19$ $$49 - 98 T + 147 T^{2} - 84 T^{3} + 35 T^{4} - 7 T^{5} + T^{6}$$
$23$ $$8281 + 4459 T + 2401 T^{2} + 182 T^{3} + 49 T^{4} + T^{6}$$
$29$ $$841 - 435 T + 283 T^{2} - 28 T^{3} + 19 T^{4} - 2 T^{5} + T^{6}$$
$31$ $$( -197 + 41 T + 16 T^{2} + T^{3} )^{2}$$
$37$ $$142129 + 59943 T + 16987 T^{2} + 2744 T^{3} + 325 T^{4} + 22 T^{5} + T^{6}$$
$41$ $$841 - 696 T + 895 T^{2} + 322 T^{3} + 97 T^{4} + 11 T^{5} + T^{6}$$
$43$ $$1681 - 1927 T + 1594 T^{2} - 623 T^{3} + 178 T^{4} - 15 T^{5} + T^{6}$$
$47$ $$( -7 + 14 T - 7 T^{2} + T^{3} )^{2}$$
$53$ $$( -41 + 66 T + 17 T^{2} + T^{3} )^{2}$$
$59$ $$10816 - 1664 T + 880 T^{2} - 112 T^{3} + 52 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$27889 - 2672 T + 2427 T^{2} - 126 T^{3} + 185 T^{4} - 13 T^{5} + T^{6}$$
$67$ $$1681 - 1886 T + 1665 T^{2} - 588 T^{3} + 167 T^{4} + 11 T^{5} + T^{6}$$
$71$ $$41209 + 18473 T + 8281 T^{2} + 406 T^{3} + 91 T^{4} + T^{6}$$
$73$ $$( 923 - 135 T - 6 T^{2} + T^{3} )^{2}$$
$79$ $$( 27 - 18 T - 3 T^{2} + T^{3} )^{2}$$
$83$ $$( -43 + 41 T - 12 T^{2} + T^{3} )^{2}$$
$89$ $$12769 + 11300 T + 10113 T^{2} + 126 T^{3} + 101 T^{4} + T^{5} + T^{6}$$
$97$ $$2679769 - 459997 T + 87146 T^{2} - 1869 T^{3} + 306 T^{4} - 5 T^{5} + T^{6}$$
show more
show less