# Properties

 Label 507.2.f.c Level $507$ Weight $2$ Character orbit 507.f Analytic conductor $4.048$ Analytic rank $0$ Dimension $4$ CM discriminant -3 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{4} + ( 2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{4} + ( 2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( -2 + 4 \zeta_{12}^{2} ) q^{12} -4 q^{16} + ( -5 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{19} + ( 5 + \zeta_{12} - \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{21} + 5 \zeta_{12}^{3} q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -2 + 6 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{28} + ( -6 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{31} + 6 \zeta_{12}^{3} q^{36} + ( -7 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{37} + ( 1 - 2 \zeta_{12}^{2} ) q^{43} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{48} + ( 3 - 6 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{49} + ( 7 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{57} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{61} + ( 6 + 9 \zeta_{12} - 9 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( -2 - 7 \zeta_{12} - 7 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{67} + ( -8 + \zeta_{12} - \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{73} + ( -5 + 10 \zeta_{12}^{2} ) q^{75} + ( 6 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{76} + ( -14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{79} + 9 q^{81} + ( 8 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{84} + ( 11 - 7 \zeta_{12} - 7 \zeta_{12}^{2} + 11 \zeta_{12}^{3} ) q^{93} + ( 8 - 11 \zeta_{12} - 11 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{7} + 12q^{9} + O(q^{10})$$ $$4q + 2q^{7} + 12q^{9} - 16q^{16} - 16q^{19} + 18q^{21} + 4q^{28} - 14q^{31} - 20q^{37} + 12q^{57} + 6q^{63} - 22q^{67} - 34q^{73} + 32q^{76} + 36q^{81} + 36q^{84} + 30q^{93} + 10q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{2}$$

## 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
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 −1.73205 2.00000i 0 0 −2.09808 2.09808i 0 3.00000 0
239.2 0 1.73205 2.00000i 0 0 3.09808 + 3.09808i 0 3.00000 0
437.1 0 −1.73205 2.00000i 0 0 −2.09808 + 2.09808i 0 3.00000 0
437.2 0 1.73205 2.00000i 0 0 3.09808 3.09808i 0 3.00000 0
 $$n$$: 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 CM by $$\Q(\sqrt{-3})$$
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.c 4
3.b odd 2 1 CM 507.2.f.c 4
13.b even 2 1 507.2.f.b 4
13.c even 3 1 39.2.k.a 4
13.c even 3 1 507.2.k.b 4
13.d odd 4 1 507.2.f.b 4
13.d odd 4 1 inner 507.2.f.c 4
13.e even 6 1 507.2.k.a 4
13.e even 6 1 507.2.k.c 4
13.f odd 12 1 39.2.k.a 4
13.f odd 12 1 507.2.k.a 4
13.f odd 12 1 507.2.k.b 4
13.f odd 12 1 507.2.k.c 4
39.d odd 2 1 507.2.f.b 4
39.f even 4 1 507.2.f.b 4
39.f even 4 1 inner 507.2.f.c 4
39.h odd 6 1 507.2.k.a 4
39.h odd 6 1 507.2.k.c 4
39.i odd 6 1 39.2.k.a 4
39.i odd 6 1 507.2.k.b 4
39.k even 12 1 39.2.k.a 4
39.k even 12 1 507.2.k.a 4
39.k even 12 1 507.2.k.b 4
39.k even 12 1 507.2.k.c 4
52.j odd 6 1 624.2.cn.b 4
52.l even 12 1 624.2.cn.b 4
65.n even 6 1 975.2.bo.c 4
65.o even 12 1 975.2.bp.a 4
65.q odd 12 1 975.2.bp.a 4
65.q odd 12 1 975.2.bp.d 4
65.s odd 12 1 975.2.bo.c 4
65.t even 12 1 975.2.bp.d 4
156.p even 6 1 624.2.cn.b 4
156.v odd 12 1 624.2.cn.b 4
195.x odd 6 1 975.2.bo.c 4
195.bc odd 12 1 975.2.bp.d 4
195.bh even 12 1 975.2.bo.c 4
195.bl even 12 1 975.2.bp.a 4
195.bl even 12 1 975.2.bp.d 4
195.bn odd 12 1 975.2.bp.a 4

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

## Hecke kernels

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

 $$T_{2}$$ $$T_{5}$$ $$T_{7}^{4} - 2 T_{7}^{3} + 2 T_{7}^{2} + 26 T_{7} + 169$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$169 + 26 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$676 + 416 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$169 - 182 T + 98 T^{2} + 14 T^{3} + T^{4}$$
$37$ $$676 + 520 T + 200 T^{2} + 20 T^{3} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 3 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -75 + T^{2} )^{2}$$
$67$ $$169 - 286 T + 242 T^{2} + 22 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$20449 + 4862 T + 578 T^{2} + 34 T^{3} + T^{4}$$
$79$ $$( -147 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$28561 + 1690 T + 50 T^{2} - 10 T^{3} + T^{4}$$