# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−1.20711 + 2.09077i −0.500000 + 0.866025i −1.91421 3.31552i −2.82843 −1.20711 2.09077i −1.41421 2.44949i 4.41421 −0.500000 0.866025i 3.41421 5.91359i
22.2 0.207107 0.358719i −0.500000 + 0.866025i 0.914214 + 1.58346i 2.82843 0.207107 + 0.358719i 1.41421 + 2.44949i 1.58579 −0.500000 0.866025i 0.585786 1.01461i
484.1 −1.20711 2.09077i −0.500000 0.866025i −1.91421 + 3.31552i −2.82843 −1.20711 + 2.09077i −1.41421 + 2.44949i 4.41421 −0.500000 + 0.866025i 3.41421 + 5.91359i
484.2 0.207107 + 0.358719i −0.500000 0.866025i 0.914214 1.58346i 2.82843 0.207107 0.358719i 1.41421 2.44949i 1.58579 −0.500000 + 0.866025i 0.585786 + 1.01461i
 $$n$$: 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.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.d 4
13.b even 2 1 507.2.e.h 4
13.c even 3 1 507.2.a.h 2
13.c even 3 1 inner 507.2.e.d 4
13.d odd 4 2 507.2.j.f 8
13.e even 6 1 39.2.a.b 2
13.e even 6 1 507.2.e.h 4
13.f odd 12 2 507.2.b.e 4
13.f odd 12 2 507.2.j.f 8
39.h odd 6 1 117.2.a.c 2
39.i odd 6 1 1521.2.a.f 2
39.k even 12 2 1521.2.b.j 4
52.i odd 6 1 624.2.a.k 2
52.j odd 6 1 8112.2.a.bm 2
65.l even 6 1 975.2.a.l 2
65.r odd 12 2 975.2.c.h 4
91.t odd 6 1 1911.2.a.h 2
104.p odd 6 1 2496.2.a.bi 2
104.s even 6 1 2496.2.a.bf 2
117.l even 6 1 1053.2.e.m 4
117.m odd 6 1 1053.2.e.e 4
117.r even 6 1 1053.2.e.m 4
117.v odd 6 1 1053.2.e.e 4
143.i odd 6 1 4719.2.a.p 2
156.r even 6 1 1872.2.a.w 2
195.y odd 6 1 2925.2.a.v 2
195.bf even 12 2 2925.2.c.u 4
273.u even 6 1 5733.2.a.u 2
312.ba even 6 1 7488.2.a.co 2
312.bg odd 6 1 7488.2.a.cl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 13.e even 6 1
117.2.a.c 2 39.h odd 6 1
507.2.a.h 2 13.c even 3 1
507.2.b.e 4 13.f odd 12 2
507.2.e.d 4 1.a even 1 1 trivial
507.2.e.d 4 13.c even 3 1 inner
507.2.e.h 4 13.b even 2 1
507.2.e.h 4 13.e even 6 1
507.2.j.f 8 13.d odd 4 2
507.2.j.f 8 13.f odd 12 2
624.2.a.k 2 52.i odd 6 1
975.2.a.l 2 65.l even 6 1
975.2.c.h 4 65.r odd 12 2
1053.2.e.e 4 117.m odd 6 1
1053.2.e.e 4 117.v odd 6 1
1053.2.e.m 4 117.l even 6 1
1053.2.e.m 4 117.r even 6 1
1521.2.a.f 2 39.i odd 6 1
1521.2.b.j 4 39.k even 12 2
1872.2.a.w 2 156.r even 6 1
1911.2.a.h 2 91.t odd 6 1
2496.2.a.bf 2 104.s even 6 1
2496.2.a.bi 2 104.p odd 6 1
2925.2.a.v 2 195.y odd 6 1
2925.2.c.u 4 195.bf even 12 2
4719.2.a.p 2 143.i odd 6 1
5733.2.a.u 2 273.u even 6 1
7488.2.a.cl 2 312.bg odd 6 1
7488.2.a.co 2 312.ba even 6 1
8112.2.a.bm 2 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{4} + 2 T_{2}^{3} + 5 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{5}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( -8 + T^{2} )^{2}$$
$7$ $$64 + 8 T^{2} + T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$64 + 8 T^{2} + T^{4}$$
$23$ $$( 16 - 4 T + T^{2} )^{2}$$
$29$ $$( 4 + 2 T + T^{2} )^{2}$$
$31$ $$( 8 - 8 T + T^{2} )^{2}$$
$37$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$
$41$ $$3136 - 896 T + 200 T^{2} - 16 T^{3} + T^{4}$$
$43$ $$256 - 128 T + 80 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$( 4 - 12 T + T^{2} )^{2}$$
$53$ $$( 2 + T )^{4}$$
$59$ $$784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$( 4 - 2 T + T^{2} )^{2}$$
$73$ $$( 4 + 12 T + T^{2} )^{2}$$
$79$ $$( -128 + T^{2} )^{2}$$
$83$ $$( -28 - 4 T + T^{2} )^{2}$$
$89$ $$18496 - 3264 T + 440 T^{2} - 24 T^{3} + T^{4}$$
$97$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$