# Properties

 Label 507.2.e.c Level $507$ Weight $2$ Character orbit 507.e Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 1.00000 + 1.73205i 3.00000 −0.500000 0.866025i 0.500000 0.866025i
484.1 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 1.73205i 3.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
 $$n$$: 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.c 2
13.b even 2 1 39.2.e.a 2
13.c even 3 1 507.2.a.b 1
13.c even 3 1 inner 507.2.e.c 2
13.d odd 4 2 507.2.j.d 4
13.e even 6 1 39.2.e.a 2
13.e even 6 1 507.2.a.c 1
13.f odd 12 2 507.2.b.b 2
13.f odd 12 2 507.2.j.d 4
39.d odd 2 1 117.2.g.b 2
39.h odd 6 1 117.2.g.b 2
39.h odd 6 1 1521.2.a.a 1
39.i odd 6 1 1521.2.a.d 1
39.k even 12 2 1521.2.b.c 2
52.b odd 2 1 624.2.q.c 2
52.i odd 6 1 624.2.q.c 2
52.i odd 6 1 8112.2.a.w 1
52.j odd 6 1 8112.2.a.bc 1
65.d even 2 1 975.2.i.f 2
65.h odd 4 2 975.2.bb.d 4
65.l even 6 1 975.2.i.f 2
65.r odd 12 2 975.2.bb.d 4
156.h even 2 1 1872.2.t.j 2
156.r even 6 1 1872.2.t.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 13.b even 2 1
39.2.e.a 2 13.e even 6 1
117.2.g.b 2 39.d odd 2 1
117.2.g.b 2 39.h odd 6 1
507.2.a.b 1 13.c even 3 1
507.2.a.c 1 13.e even 6 1
507.2.b.b 2 13.f odd 12 2
507.2.e.c 2 1.a even 1 1 trivial
507.2.e.c 2 13.c even 3 1 inner
507.2.j.d 4 13.d odd 4 2
507.2.j.d 4 13.f odd 12 2
624.2.q.c 2 52.b odd 2 1
624.2.q.c 2 52.i odd 6 1
975.2.i.f 2 65.d even 2 1
975.2.i.f 2 65.l even 6 1
975.2.bb.d 4 65.h odd 4 2
975.2.bb.d 4 65.r odd 12 2
1521.2.a.a 1 39.h odd 6 1
1521.2.a.d 1 39.i odd 6 1
1521.2.b.c 2 39.k even 12 2
1872.2.t.j 2 156.h even 2 1
1872.2.t.j 2 156.r even 6 1
8112.2.a.w 1 52.i odd 6 1
8112.2.a.bc 1 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{5} - 1$$ T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 7T + 49$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - T + 1$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} + 6T + 36$$
$47$ $$(T + 6)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$(T + 11)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 14)^{2}$$
$89$ $$T^{2} + 14T + 196$$
$97$ $$T^{2} + 2T + 4$$