# Properties

 Label 507.2.j.a Level $507$ Weight $2$ Character orbit 507.j 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.j (of order $$6$$, 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + (\zeta_{6} - 2) q^{6} + (2 \zeta_{6} - 4) q^{7} + (2 \zeta_{6} - 1) q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z - 1) * q^2 + (-z + 1) * q^3 + z * q^4 + (z - 2) * q^6 + (2*z - 4) * q^7 + (2*z - 1) * q^8 - z * q^9 $$q + ( - \zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + (\zeta_{6} - 2) q^{6} + (2 \zeta_{6} - 4) q^{7} + (2 \zeta_{6} - 1) q^{8} - \zeta_{6} q^{9} + (2 \zeta_{6} + 2) q^{11} + q^{12} + 6 q^{14} + ( - 5 \zeta_{6} + 5) q^{16} + 6 \zeta_{6} q^{17} + (2 \zeta_{6} - 1) q^{18} + ( - 2 \zeta_{6} + 4) q^{19} + (4 \zeta_{6} - 2) q^{21} - 6 \zeta_{6} q^{22} + (\zeta_{6} + 1) q^{24} + 5 q^{25} - q^{27} + ( - 2 \zeta_{6} - 2) q^{28} + (6 \zeta_{6} - 6) q^{29} + ( - 4 \zeta_{6} + 2) q^{31} + (3 \zeta_{6} - 6) q^{32} + ( - 2 \zeta_{6} + 4) q^{33} + ( - 12 \zeta_{6} + 6) q^{34} + ( - \zeta_{6} + 1) q^{36} + ( - 4 \zeta_{6} - 4) q^{37} - 6 q^{38} + (4 \zeta_{6} + 4) q^{41} + ( - 6 \zeta_{6} + 6) q^{42} + 4 \zeta_{6} q^{43} + (4 \zeta_{6} - 2) q^{44} + (4 \zeta_{6} - 2) q^{47} - 5 \zeta_{6} q^{48} + ( - 5 \zeta_{6} + 5) q^{49} + ( - 5 \zeta_{6} - 5) q^{50} + 6 q^{51} + 6 q^{53} + (\zeta_{6} + 1) q^{54} - 6 \zeta_{6} q^{56} + ( - 4 \zeta_{6} + 2) q^{57} + ( - 6 \zeta_{6} + 12) q^{58} + ( - 6 \zeta_{6} + 12) q^{59} + 2 \zeta_{6} q^{61} + (6 \zeta_{6} - 6) q^{62} + (2 \zeta_{6} + 2) q^{63} - q^{64} - 6 q^{66} + (6 \zeta_{6} + 6) q^{67} + (6 \zeta_{6} - 6) q^{68} + ( - 2 \zeta_{6} + 4) q^{71} + ( - \zeta_{6} + 2) q^{72} + 12 \zeta_{6} q^{74} + ( - 5 \zeta_{6} + 5) q^{75} + (2 \zeta_{6} + 2) q^{76} - 12 q^{77} - 8 q^{79} + (\zeta_{6} - 1) q^{81} - 12 \zeta_{6} q^{82} + ( - 4 \zeta_{6} + 2) q^{83} + (2 \zeta_{6} - 4) q^{84} + ( - 8 \zeta_{6} + 4) q^{86} + 6 \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{88} + (4 \zeta_{6} + 4) q^{89} + ( - 2 \zeta_{6} - 2) q^{93} + ( - 6 \zeta_{6} + 6) q^{94} + (6 \zeta_{6} - 3) q^{96} + (8 \zeta_{6} - 16) q^{97} + (5 \zeta_{6} - 10) q^{98} + ( - 4 \zeta_{6} + 2) q^{99} +O(q^{100})$$ q + (-z - 1) * q^2 + (-z + 1) * q^3 + z * q^4 + (z - 2) * q^6 + (2*z - 4) * q^7 + (2*z - 1) * q^8 - z * q^9 + (2*z + 2) * q^11 + q^12 + 6 * q^14 + (-5*z + 5) * q^16 + 6*z * q^17 + (2*z - 1) * q^18 + (-2*z + 4) * q^19 + (4*z - 2) * q^21 - 6*z * q^22 + (z + 1) * q^24 + 5 * q^25 - q^27 + (-2*z - 2) * q^28 + (6*z - 6) * q^29 + (-4*z + 2) * q^31 + (3*z - 6) * q^32 + (-2*z + 4) * q^33 + (-12*z + 6) * q^34 + (-z + 1) * q^36 + (-4*z - 4) * q^37 - 6 * q^38 + (4*z + 4) * q^41 + (-6*z + 6) * q^42 + 4*z * q^43 + (4*z - 2) * q^44 + (4*z - 2) * q^47 - 5*z * q^48 + (-5*z + 5) * q^49 + (-5*z - 5) * q^50 + 6 * q^51 + 6 * q^53 + (z + 1) * q^54 - 6*z * q^56 + (-4*z + 2) * q^57 + (-6*z + 12) * q^58 + (-6*z + 12) * q^59 + 2*z * q^61 + (6*z - 6) * q^62 + (2*z + 2) * q^63 - q^64 - 6 * q^66 + (6*z + 6) * q^67 + (6*z - 6) * q^68 + (-2*z + 4) * q^71 + (-z + 2) * q^72 + 12*z * q^74 + (-5*z + 5) * q^75 + (2*z + 2) * q^76 - 12 * q^77 - 8 * q^79 + (z - 1) * q^81 - 12*z * q^82 + (-4*z + 2) * q^83 + (2*z - 4) * q^84 + (-8*z + 4) * q^86 + 6*z * q^87 + (6*z - 6) * q^88 + (4*z + 4) * q^89 + (-2*z - 2) * q^93 + (-6*z + 6) * q^94 + (6*z - 3) * q^96 + (8*z - 16) * q^97 + (5*z - 10) * q^98 + (-4*z + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + q^{3} + q^{4} - 3 q^{6} - 6 q^{7} - q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + q^3 + q^4 - 3 * q^6 - 6 * q^7 - q^9 $$2 q - 3 q^{2} + q^{3} + q^{4} - 3 q^{6} - 6 q^{7} - q^{9} + 6 q^{11} + 2 q^{12} + 12 q^{14} + 5 q^{16} + 6 q^{17} + 6 q^{19} - 6 q^{22} + 3 q^{24} + 10 q^{25} - 2 q^{27} - 6 q^{28} - 6 q^{29} - 9 q^{32} + 6 q^{33} + q^{36} - 12 q^{37} - 12 q^{38} + 12 q^{41} + 6 q^{42} + 4 q^{43} - 5 q^{48} + 5 q^{49} - 15 q^{50} + 12 q^{51} + 12 q^{53} + 3 q^{54} - 6 q^{56} + 18 q^{58} + 18 q^{59} + 2 q^{61} - 6 q^{62} + 6 q^{63} - 2 q^{64} - 12 q^{66} + 18 q^{67} - 6 q^{68} + 6 q^{71} + 3 q^{72} + 12 q^{74} + 5 q^{75} + 6 q^{76} - 24 q^{77} - 16 q^{79} - q^{81} - 12 q^{82} - 6 q^{84} + 6 q^{87} - 6 q^{88} + 12 q^{89} - 6 q^{93} + 6 q^{94} - 24 q^{97} - 15 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 + q^3 + q^4 - 3 * q^6 - 6 * q^7 - q^9 + 6 * q^11 + 2 * q^12 + 12 * q^14 + 5 * q^16 + 6 * q^17 + 6 * q^19 - 6 * q^22 + 3 * q^24 + 10 * q^25 - 2 * q^27 - 6 * q^28 - 6 * q^29 - 9 * q^32 + 6 * q^33 + q^36 - 12 * q^37 - 12 * q^38 + 12 * q^41 + 6 * q^42 + 4 * q^43 - 5 * q^48 + 5 * q^49 - 15 * q^50 + 12 * q^51 + 12 * q^53 + 3 * q^54 - 6 * q^56 + 18 * q^58 + 18 * q^59 + 2 * q^61 - 6 * q^62 + 6 * q^63 - 2 * q^64 - 12 * q^66 + 18 * q^67 - 6 * q^68 + 6 * q^71 + 3 * q^72 + 12 * q^74 + 5 * q^75 + 6 * q^76 - 24 * q^77 - 16 * q^79 - q^81 - 12 * q^82 - 6 * q^84 + 6 * q^87 - 6 * q^88 + 12 * q^89 - 6 * q^93 + 6 * q^94 - 24 * q^97 - 15 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 0 −1.50000 + 0.866025i −3.00000 + 1.73205i 1.73205i −0.500000 0.866025i 0
361.1 −1.50000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 0 −1.50000 0.866025i −3.00000 1.73205i 1.73205i −0.500000 + 0.866025i 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.a 2
13.b even 2 1 507.2.j.c 2
13.c even 3 1 39.2.b.a 2
13.c even 3 1 507.2.j.c 2
13.d odd 4 2 507.2.e.e 4
13.e even 6 1 39.2.b.a 2
13.e even 6 1 inner 507.2.j.a 2
13.f odd 12 2 507.2.a.f 2
13.f odd 12 2 507.2.e.e 4
39.h odd 6 1 117.2.b.a 2
39.i odd 6 1 117.2.b.a 2
39.k even 12 2 1521.2.a.l 2
52.i odd 6 1 624.2.c.e 2
52.j odd 6 1 624.2.c.e 2
52.l even 12 2 8112.2.a.bv 2
65.l even 6 1 975.2.b.d 2
65.n even 6 1 975.2.b.d 2
65.q odd 12 2 975.2.h.f 4
65.r odd 12 2 975.2.h.f 4
91.n odd 6 1 1911.2.c.d 2
91.t odd 6 1 1911.2.c.d 2
104.n odd 6 1 2496.2.c.d 2
104.p odd 6 1 2496.2.c.d 2
104.r even 6 1 2496.2.c.k 2
104.s even 6 1 2496.2.c.k 2
156.p even 6 1 1872.2.c.e 2
156.r even 6 1 1872.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 13.c even 3 1
39.2.b.a 2 13.e even 6 1
117.2.b.a 2 39.h odd 6 1
117.2.b.a 2 39.i odd 6 1
507.2.a.f 2 13.f odd 12 2
507.2.e.e 4 13.d odd 4 2
507.2.e.e 4 13.f odd 12 2
507.2.j.a 2 1.a even 1 1 trivial
507.2.j.a 2 13.e even 6 1 inner
507.2.j.c 2 13.b even 2 1
507.2.j.c 2 13.c even 3 1
624.2.c.e 2 52.i odd 6 1
624.2.c.e 2 52.j odd 6 1
975.2.b.d 2 65.l even 6 1
975.2.b.d 2 65.n even 6 1
975.2.h.f 4 65.q odd 12 2
975.2.h.f 4 65.r odd 12 2
1521.2.a.l 2 39.k even 12 2
1872.2.c.e 2 156.p even 6 1
1872.2.c.e 2 156.r even 6 1
1911.2.c.d 2 91.n odd 6 1
1911.2.c.d 2 91.t odd 6 1
2496.2.c.d 2 104.n odd 6 1
2496.2.c.d 2 104.p odd 6 1
2496.2.c.k 2 104.r even 6 1
2496.2.c.k 2 104.s even 6 1
8112.2.a.bv 2 52.l even 12 2

## Hecke kernels

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

 $$T_{2}^{2} + 3T_{2} + 3$$ T2^2 + 3*T2 + 3 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 3$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 12$$
$11$ $$T^{2} - 6T + 12$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 6T + 12$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} + 12T + 48$$
$41$ $$T^{2} - 12T + 48$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} + 12$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 18T + 108$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} - 18T + 108$$
$71$ $$T^{2} - 6T + 12$$
$73$ $$T^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} - 12T + 48$$
$97$ $$T^{2} + 24T + 192$$