# Properties

 Label 507.2.k.i Level $507$ Weight $2$ Character orbit 507.k Analytic conductor $4.048$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.k (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i
−0.258819 + 0.965926i 1.72474 + 0.158919i 0.866025 + 0.500000i 1.41421 + 1.41421i −0.599900 + 1.62484i 1.36603 0.366025i −2.12132 + 2.12132i 2.94949 + 0.548188i −1.73205 + 1.00000i
80.2 0.258819 0.965926i −0.724745 + 1.57313i 0.866025 + 0.500000i −1.41421 1.41421i 1.33195 + 1.10721i 1.36603 0.366025i 2.12132 2.12132i −1.94949 2.28024i −1.73205 + 1.00000i
89.1 −0.965926 0.258819i 1.72474 + 0.158919i −0.866025 0.500000i −1.41421 + 1.41421i −1.62484 0.599900i −0.366025 1.36603i 2.12132 + 2.12132i 2.94949 + 0.548188i 1.73205 1.00000i
89.2 0.965926 + 0.258819i −0.724745 + 1.57313i −0.866025 0.500000i 1.41421 1.41421i −1.10721 + 1.33195i −0.366025 1.36603i −2.12132 2.12132i −1.94949 2.28024i 1.73205 1.00000i
188.1 −0.965926 + 0.258819i 1.72474 0.158919i −0.866025 + 0.500000i −1.41421 1.41421i −1.62484 + 0.599900i −0.366025 + 1.36603i 2.12132 2.12132i 2.94949 0.548188i 1.73205 + 1.00000i
188.2 0.965926 0.258819i −0.724745 1.57313i −0.866025 + 0.500000i 1.41421 + 1.41421i −1.10721 1.33195i −0.366025 + 1.36603i −2.12132 + 2.12132i −1.94949 + 2.28024i 1.73205 + 1.00000i
488.1 −0.258819 0.965926i 1.72474 0.158919i 0.866025 0.500000i 1.41421 1.41421i −0.599900 1.62484i 1.36603 + 0.366025i −2.12132 2.12132i 2.94949 0.548188i −1.73205 1.00000i
488.2 0.258819 + 0.965926i −0.724745 1.57313i 0.866025 0.500000i −1.41421 + 1.41421i 1.33195 1.10721i 1.36603 + 0.366025i 2.12132 + 2.12132i −1.94949 + 2.28024i −1.73205 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 488.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner
39.f even 4 1 inner
39.i odd 6 1 inner
39.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.k.i 8
3.b odd 2 1 inner 507.2.k.i 8
13.b even 2 1 507.2.k.j 8
13.c even 3 1 507.2.f.a 4
13.c even 3 1 inner 507.2.k.i 8
13.d odd 4 1 inner 507.2.k.i 8
13.d odd 4 1 507.2.k.j 8
13.e even 6 1 39.2.f.a 4
13.e even 6 1 507.2.k.j 8
13.f odd 12 1 39.2.f.a 4
13.f odd 12 1 507.2.f.a 4
13.f odd 12 1 inner 507.2.k.i 8
13.f odd 12 1 507.2.k.j 8
39.d odd 2 1 507.2.k.j 8
39.f even 4 1 inner 507.2.k.i 8
39.f even 4 1 507.2.k.j 8
39.h odd 6 1 39.2.f.a 4
39.h odd 6 1 507.2.k.j 8
39.i odd 6 1 507.2.f.a 4
39.i odd 6 1 inner 507.2.k.i 8
39.k even 12 1 39.2.f.a 4
39.k even 12 1 507.2.f.a 4
39.k even 12 1 inner 507.2.k.i 8
39.k even 12 1 507.2.k.j 8
52.i odd 6 1 624.2.bf.d 4
52.l even 12 1 624.2.bf.d 4
65.l even 6 1 975.2.o.j 4
65.o even 12 1 975.2.n.d 4
65.r odd 12 1 975.2.n.c 4
65.r odd 12 1 975.2.n.d 4
65.s odd 12 1 975.2.o.j 4
65.t even 12 1 975.2.n.c 4
156.r even 6 1 624.2.bf.d 4
156.v odd 12 1 624.2.bf.d 4
195.y odd 6 1 975.2.o.j 4
195.bc odd 12 1 975.2.n.c 4
195.bf even 12 1 975.2.n.c 4
195.bf even 12 1 975.2.n.d 4
195.bh even 12 1 975.2.o.j 4
195.bn odd 12 1 975.2.n.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.f.a 4 13.e even 6 1
39.2.f.a 4 13.f odd 12 1
39.2.f.a 4 39.h odd 6 1
39.2.f.a 4 39.k even 12 1
507.2.f.a 4 13.c even 3 1
507.2.f.a 4 13.f odd 12 1
507.2.f.a 4 39.i odd 6 1
507.2.f.a 4 39.k even 12 1
507.2.k.i 8 1.a even 1 1 trivial
507.2.k.i 8 3.b odd 2 1 inner
507.2.k.i 8 13.c even 3 1 inner
507.2.k.i 8 13.d odd 4 1 inner
507.2.k.i 8 13.f odd 12 1 inner
507.2.k.i 8 39.f even 4 1 inner
507.2.k.i 8 39.i odd 6 1 inner
507.2.k.i 8 39.k even 12 1 inner
507.2.k.j 8 13.b even 2 1
507.2.k.j 8 13.d odd 4 1
507.2.k.j 8 13.e even 6 1
507.2.k.j 8 13.f odd 12 1
507.2.k.j 8 39.d odd 2 1
507.2.k.j 8 39.f even 4 1
507.2.k.j 8 39.h odd 6 1
507.2.k.j 8 39.k even 12 1
624.2.bf.d 4 52.i odd 6 1
624.2.bf.d 4 52.l even 12 1
624.2.bf.d 4 156.r even 6 1
624.2.bf.d 4 156.v odd 12 1
975.2.n.c 4 65.r odd 12 1
975.2.n.c 4 65.t even 12 1
975.2.n.c 4 195.bc odd 12 1
975.2.n.c 4 195.bf even 12 1
975.2.n.d 4 65.o even 12 1
975.2.n.d 4 65.r odd 12 1
975.2.n.d 4 195.bf even 12 1
975.2.n.d 4 195.bn odd 12 1
975.2.o.j 4 65.l even 6 1
975.2.o.j 4 65.s odd 12 1
975.2.o.j 4 195.y odd 6 1
975.2.o.j 4 195.bh even 12 1

## Hecke kernels

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

 $$T_{2}^{8} - T_{2}^{4} + 1$$ $$T_{5}^{4} + 16$$ $$T_{7}^{4} - 2 T_{7}^{3} + 2 T_{7}^{2} - 4 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$( 9 - 6 T + T^{2} - 2 T^{3} + T^{4} )^{2}$$
$5$ $$( 16 + T^{4} )^{2}$$
$7$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$65536 - 256 T^{4} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$23$ $$( 5184 + 72 T^{2} + T^{4} )^{2}$$
$29$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$31$ $$( 50 - 10 T + T^{2} )^{4}$$
$37$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$41$ $$256 - 16 T^{4} + T^{8}$$
$43$ $$( 1296 - 36 T^{2} + T^{4} )^{2}$$
$47$ $$( 256 + T^{4} )^{2}$$
$53$ $$( 32 + T^{2} )^{4}$$
$59$ $$65536 - 256 T^{4} + T^{8}$$
$61$ $$( 64 + 8 T + T^{2} )^{4}$$
$67$ $$( 2500 + 500 T + 50 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$71$ $$65536 - 256 T^{4} + T^{8}$$
$73$ $$( 2 + 2 T + T^{2} )^{4}$$
$79$ $$( 10 + T )^{8}$$
$83$ $$( 4096 + T^{4} )^{2}$$
$89$ $$1475789056 - 38416 T^{4} + T^{8}$$
$97$ $$( 9604 - 1372 T + 98 T^{2} - 14 T^{3} + T^{4} )^{2}$$