# Properties

 Label 507.2.e.a 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + 2 q^{5} + \zeta_{6} q^{6} + 4 \zeta_{6} q^{7} - 3 q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 + z * q^4 + 2 * q^5 + z * q^6 + 4*z * q^7 - 3 * q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + 2 q^{5} + \zeta_{6} q^{6} + 4 \zeta_{6} q^{7} - 3 q^{8} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + (4 \zeta_{6} - 4) q^{11} + q^{12} - 4 q^{14} + ( - 2 \zeta_{6} + 2) q^{15} + ( - \zeta_{6} + 1) q^{16} - 2 \zeta_{6} q^{17} + q^{18} + 2 \zeta_{6} q^{20} + 4 q^{21} - 4 \zeta_{6} q^{22} + (3 \zeta_{6} - 3) q^{24} - q^{25} - q^{27} + (4 \zeta_{6} - 4) q^{28} + ( - 10 \zeta_{6} + 10) q^{29} + 2 \zeta_{6} q^{30} + 4 q^{31} - 5 \zeta_{6} q^{32} + 4 \zeta_{6} q^{33} + 2 q^{34} + 8 \zeta_{6} q^{35} + ( - \zeta_{6} + 1) q^{36} + ( - 2 \zeta_{6} + 2) q^{37} - 6 q^{40} + (6 \zeta_{6} - 6) q^{41} + (4 \zeta_{6} - 4) q^{42} + 12 \zeta_{6} q^{43} - 4 q^{44} - 2 \zeta_{6} q^{45} - \zeta_{6} q^{48} + (9 \zeta_{6} - 9) q^{49} + ( - \zeta_{6} + 1) q^{50} - 2 q^{51} + 6 q^{53} + ( - \zeta_{6} + 1) q^{54} + (8 \zeta_{6} - 8) q^{55} - 12 \zeta_{6} q^{56} + 10 \zeta_{6} q^{58} - 12 \zeta_{6} q^{59} + 2 q^{60} + 2 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + ( - 4 \zeta_{6} + 4) q^{63} + 7 q^{64} - 4 q^{66} + ( - 8 \zeta_{6} + 8) q^{67} + ( - 2 \zeta_{6} + 2) q^{68} - 8 q^{70} + 3 \zeta_{6} q^{72} + 2 q^{73} + 2 \zeta_{6} q^{74} + (\zeta_{6} - 1) q^{75} - 16 q^{77} + 8 q^{79} + ( - 2 \zeta_{6} + 2) q^{80} + (\zeta_{6} - 1) q^{81} - 6 \zeta_{6} q^{82} + 4 q^{83} + 4 \zeta_{6} q^{84} - 4 \zeta_{6} q^{85} - 12 q^{86} - 10 \zeta_{6} q^{87} + ( - 12 \zeta_{6} + 12) q^{88} + ( - 2 \zeta_{6} + 2) q^{89} + 2 q^{90} + ( - 4 \zeta_{6} + 4) q^{93} - 5 q^{96} - 10 \zeta_{6} q^{97} - 9 \zeta_{6} q^{98} + 4 q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 + z * q^4 + 2 * q^5 + z * q^6 + 4*z * q^7 - 3 * q^8 - z * q^9 + (2*z - 2) * q^10 + (4*z - 4) * q^11 + q^12 - 4 * q^14 + (-2*z + 2) * q^15 + (-z + 1) * q^16 - 2*z * q^17 + q^18 + 2*z * q^20 + 4 * q^21 - 4*z * q^22 + (3*z - 3) * q^24 - q^25 - q^27 + (4*z - 4) * q^28 + (-10*z + 10) * q^29 + 2*z * q^30 + 4 * q^31 - 5*z * q^32 + 4*z * q^33 + 2 * q^34 + 8*z * q^35 + (-z + 1) * q^36 + (-2*z + 2) * q^37 - 6 * q^40 + (6*z - 6) * q^41 + (4*z - 4) * q^42 + 12*z * q^43 - 4 * q^44 - 2*z * q^45 - z * q^48 + (9*z - 9) * q^49 + (-z + 1) * q^50 - 2 * q^51 + 6 * q^53 + (-z + 1) * q^54 + (8*z - 8) * q^55 - 12*z * q^56 + 10*z * q^58 - 12*z * q^59 + 2 * q^60 + 2*z * q^61 + (4*z - 4) * q^62 + (-4*z + 4) * q^63 + 7 * q^64 - 4 * q^66 + (-8*z + 8) * q^67 + (-2*z + 2) * q^68 - 8 * q^70 + 3*z * q^72 + 2 * q^73 + 2*z * q^74 + (z - 1) * q^75 - 16 * q^77 + 8 * q^79 + (-2*z + 2) * q^80 + (z - 1) * q^81 - 6*z * q^82 + 4 * q^83 + 4*z * q^84 - 4*z * q^85 - 12 * q^86 - 10*z * q^87 + (-12*z + 12) * q^88 + (-2*z + 2) * q^89 + 2 * q^90 + (-4*z + 4) * q^93 - 5 * q^96 - 10*z * q^97 - 9*z * q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} + q^{4} + 4 q^{5} + q^{6} + 4 q^{7} - 6 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 + q^3 + q^4 + 4 * q^5 + q^6 + 4 * q^7 - 6 * q^8 - q^9 $$2 q - q^{2} + q^{3} + q^{4} + 4 q^{5} + q^{6} + 4 q^{7} - 6 q^{8} - q^{9} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 8 q^{14} + 2 q^{15} + q^{16} - 2 q^{17} + 2 q^{18} + 2 q^{20} + 8 q^{21} - 4 q^{22} - 3 q^{24} - 2 q^{25} - 2 q^{27} - 4 q^{28} + 10 q^{29} + 2 q^{30} + 8 q^{31} - 5 q^{32} + 4 q^{33} + 4 q^{34} + 8 q^{35} + q^{36} + 2 q^{37} - 12 q^{40} - 6 q^{41} - 4 q^{42} + 12 q^{43} - 8 q^{44} - 2 q^{45} - q^{48} - 9 q^{49} + q^{50} - 4 q^{51} + 12 q^{53} + q^{54} - 8 q^{55} - 12 q^{56} + 10 q^{58} - 12 q^{59} + 4 q^{60} + 2 q^{61} - 4 q^{62} + 4 q^{63} + 14 q^{64} - 8 q^{66} + 8 q^{67} + 2 q^{68} - 16 q^{70} + 3 q^{72} + 4 q^{73} + 2 q^{74} - q^{75} - 32 q^{77} + 16 q^{79} + 2 q^{80} - q^{81} - 6 q^{82} + 8 q^{83} + 4 q^{84} - 4 q^{85} - 24 q^{86} - 10 q^{87} + 12 q^{88} + 2 q^{89} + 4 q^{90} + 4 q^{93} - 10 q^{96} - 10 q^{97} - 9 q^{98} + 8 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^3 + q^4 + 4 * q^5 + q^6 + 4 * q^7 - 6 * q^8 - q^9 - 2 * q^10 - 4 * q^11 + 2 * q^12 - 8 * q^14 + 2 * q^15 + q^16 - 2 * q^17 + 2 * q^18 + 2 * q^20 + 8 * q^21 - 4 * q^22 - 3 * q^24 - 2 * q^25 - 2 * q^27 - 4 * q^28 + 10 * q^29 + 2 * q^30 + 8 * q^31 - 5 * q^32 + 4 * q^33 + 4 * q^34 + 8 * q^35 + q^36 + 2 * q^37 - 12 * q^40 - 6 * q^41 - 4 * q^42 + 12 * q^43 - 8 * q^44 - 2 * q^45 - q^48 - 9 * q^49 + q^50 - 4 * q^51 + 12 * q^53 + q^54 - 8 * q^55 - 12 * q^56 + 10 * q^58 - 12 * q^59 + 4 * q^60 + 2 * q^61 - 4 * q^62 + 4 * q^63 + 14 * q^64 - 8 * q^66 + 8 * q^67 + 2 * q^68 - 16 * q^70 + 3 * q^72 + 4 * q^73 + 2 * q^74 - q^75 - 32 * q^77 + 16 * q^79 + 2 * q^80 - q^81 - 6 * q^82 + 8 * q^83 + 4 * q^84 - 4 * q^85 - 24 * q^86 - 10 * q^87 + 12 * q^88 + 2 * q^89 + 4 * q^90 + 4 * q^93 - 10 * q^96 - 10 * q^97 - 9 * q^98 + 8 * 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 2.00000 0.500000 + 0.866025i 2.00000 + 3.46410i −3.00000 −0.500000 0.866025i −1.00000 + 1.73205i
484.1 −0.500000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 2.00000 0.500000 0.866025i 2.00000 3.46410i −3.00000 −0.500000 + 0.866025i −1.00000 1.73205i
 $$n$$: 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.a 2
13.b even 2 1 507.2.e.b 2
13.c even 3 1 39.2.a.a 1
13.c even 3 1 inner 507.2.e.a 2
13.d odd 4 2 507.2.j.e 4
13.e even 6 1 507.2.a.a 1
13.e even 6 1 507.2.e.b 2
13.f odd 12 2 507.2.b.a 2
13.f odd 12 2 507.2.j.e 4
39.h odd 6 1 1521.2.a.e 1
39.i odd 6 1 117.2.a.a 1
39.k even 12 2 1521.2.b.b 2
52.i odd 6 1 8112.2.a.s 1
52.j odd 6 1 624.2.a.i 1
65.n even 6 1 975.2.a.f 1
65.q odd 12 2 975.2.c.f 2
91.n odd 6 1 1911.2.a.f 1
104.n odd 6 1 2496.2.a.e 1
104.r even 6 1 2496.2.a.q 1
117.f even 3 1 1053.2.e.b 2
117.h even 3 1 1053.2.e.b 2
117.k odd 6 1 1053.2.e.d 2
117.u odd 6 1 1053.2.e.d 2
143.k odd 6 1 4719.2.a.c 1
156.p even 6 1 1872.2.a.h 1
195.x odd 6 1 2925.2.a.p 1
195.bl even 12 2 2925.2.c.e 2
273.bn even 6 1 5733.2.a.e 1
312.bh odd 6 1 7488.2.a.bl 1
312.bn even 6 1 7488.2.a.by 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 13.c even 3 1
117.2.a.a 1 39.i odd 6 1
507.2.a.a 1 13.e even 6 1
507.2.b.a 2 13.f odd 12 2
507.2.e.a 2 1.a even 1 1 trivial
507.2.e.a 2 13.c even 3 1 inner
507.2.e.b 2 13.b even 2 1
507.2.e.b 2 13.e even 6 1
507.2.j.e 4 13.d odd 4 2
507.2.j.e 4 13.f odd 12 2
624.2.a.i 1 52.j odd 6 1
975.2.a.f 1 65.n even 6 1
975.2.c.f 2 65.q odd 12 2
1053.2.e.b 2 117.f even 3 1
1053.2.e.b 2 117.h even 3 1
1053.2.e.d 2 117.k odd 6 1
1053.2.e.d 2 117.u odd 6 1
1521.2.a.e 1 39.h odd 6 1
1521.2.b.b 2 39.k even 12 2
1872.2.a.h 1 156.p even 6 1
1911.2.a.f 1 91.n odd 6 1
2496.2.a.e 1 104.n odd 6 1
2496.2.a.q 1 104.r even 6 1
2925.2.a.p 1 195.x odd 6 1
2925.2.c.e 2 195.bl even 12 2
4719.2.a.c 1 143.k odd 6 1
5733.2.a.e 1 273.bn even 6 1
7488.2.a.bl 1 312.bh odd 6 1
7488.2.a.by 1 312.bn even 6 1
8112.2.a.s 1 52.i 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 $$T_{5} - 2$$ T5 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 10T + 100$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} - 2T + 4$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 12T + 144$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} - 8T + 64$$
$71$ $$T^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 2T + 4$$
$97$ $$T^{2} + 10T + 100$$