# Properties

 Label 507.2.f.a Level $507$ Weight $2$ Character orbit 507.f Analytic conductor $4.048$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 0.707107i −1.00000 + 1.41421i 1.00000i 1.41421 + 1.41421i 1.70711 0.292893i −1.00000 1.00000i −2.12132 + 2.12132i −1.00000 2.82843i 2.00000i
239.2 0.707107 + 0.707107i −1.00000 1.41421i 1.00000i −1.41421 1.41421i 0.292893 1.70711i −1.00000 1.00000i 2.12132 2.12132i −1.00000 + 2.82843i 2.00000i
437.1 −0.707107 + 0.707107i −1.00000 1.41421i 1.00000i 1.41421 1.41421i 1.70711 + 0.292893i −1.00000 + 1.00000i −2.12132 2.12132i −1.00000 + 2.82843i 2.00000i
437.2 0.707107 0.707107i −1.00000 + 1.41421i 1.00000i −1.41421 + 1.41421i 0.292893 + 1.70711i −1.00000 + 1.00000i 2.12132 + 2.12132i −1.00000 2.82843i 2.00000i
 $$n$$: 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
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

## Twists

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

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

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