# Properties

 Label 507.2.b.f 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

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
 − 0.445042i − 1.80194i − 1.24698i 1.24698i 1.80194i 0.445042i
2.69202i −1.00000 −5.24698 1.04892i 2.69202i 0.554958i 8.74094i 1.00000 2.82371
337.2 2.35690i −1.00000 −3.55496 3.69202i 2.35690i 0.801938i 3.66487i 1.00000 −8.70171
337.3 2.04892i −1.00000 −2.19806 3.35690i 2.04892i 2.24698i 0.405813i 1.00000 6.87800
337.4 2.04892i −1.00000 −2.19806 3.35690i 2.04892i 2.24698i 0.405813i 1.00000 6.87800
337.5 2.35690i −1.00000 −3.55496 3.69202i 2.35690i 0.801938i 3.66487i 1.00000 −8.70171
337.6 2.69202i −1.00000 −5.24698 1.04892i 2.69202i 0.554958i 8.74094i 1.00000 2.82371
 $$n$$: 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.f 6
3.b odd 2 1 1521.2.b.k 6
13.b even 2 1 inner 507.2.b.f 6
13.c even 3 2 507.2.j.i 12
13.d odd 4 1 507.2.a.i 3
13.d odd 4 1 507.2.a.l yes 3
13.e even 6 2 507.2.j.i 12
13.f odd 12 2 507.2.e.i 6
13.f odd 12 2 507.2.e.l 6
39.d odd 2 1 1521.2.b.k 6
39.f even 4 1 1521.2.a.n 3
39.f even 4 1 1521.2.a.s 3
52.f even 4 1 8112.2.a.cg 3
52.f even 4 1 8112.2.a.cp 3

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

 $$T_{2}^{6} + 17 T_{2}^{4} + 94 T_{2}^{2} + 169$$ $$T_{5}^{6} + 26 T_{5}^{4} + 181 T_{5}^{2} + 169$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$169 + 94 T^{2} + 17 T^{4} + T^{6}$$
$3$ $$( 1 + T )^{6}$$
$5$ $$169 + 181 T^{2} + 26 T^{4} + T^{6}$$
$7$ $$1 + 5 T^{2} + 6 T^{4} + T^{6}$$
$11$ $$1681 + 474 T^{2} + 41 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( -13 - 16 T - T^{2} + T^{3} )^{2}$$
$19$ $$49 + 98 T^{2} + 21 T^{4} + T^{6}$$
$23$ $$( -91 - 49 T + T^{3} )^{2}$$
$29$ $$( -29 - 15 T + 2 T^{2} + T^{3} )^{2}$$
$31$ $$38809 + 7985 T^{2} + 174 T^{4} + T^{6}$$
$37$ $$142129 + 8693 T^{2} + 166 T^{4} + T^{6}$$
$41$ $$841 + 1214 T^{2} + 73 T^{4} + T^{6}$$
$43$ $$( -41 + 47 T - 15 T^{2} + T^{3} )^{2}$$
$47$ $$49 + 98 T^{2} + 21 T^{4} + T^{6}$$
$53$ $$( -41 + 66 T + 17 T^{2} + T^{3} )^{2}$$
$59$ $$10816 + 1504 T^{2} + 68 T^{4} + T^{6}$$
$61$ $$( -167 - 16 T + 13 T^{2} + T^{3} )^{2}$$
$67$ $$1681 + 1214 T^{2} + 213 T^{4} + T^{6}$$
$71$ $$41209 + 8281 T^{2} + 182 T^{4} + T^{6}$$
$73$ $$851929 + 29301 T^{2} + 306 T^{4} + T^{6}$$
$79$ $$( 27 - 18 T - 3 T^{2} + T^{3} )^{2}$$
$83$ $$1849 + 649 T^{2} + 62 T^{4} + T^{6}$$
$89$ $$12769 + 10226 T^{2} + 201 T^{4} + T^{6}$$
$97$ $$2679769 + 95331 T^{2} + 587 T^{4} + T^{6}$$