# Properties

 Label 507.2.j.d Level $507$ Weight $2$ Character orbit 507.j 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.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.500000 0.866025i −0.500000 0.866025i 1.00000i −0.866025 + 0.500000i −1.73205 + 1.00000i 3.00000i −0.500000 0.866025i −0.500000 + 0.866025i
316.2 0.866025 + 0.500000i 0.500000 0.866025i −0.500000 0.866025i 1.00000i 0.866025 0.500000i 1.73205 1.00000i 3.00000i −0.500000 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i −0.866025 0.500000i −1.73205 1.00000i 3.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.2 0.866025 0.500000i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i 0.866025 + 0.500000i 1.73205 + 1.00000i 3.00000i −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.b even 2 1 inner
13.c even 3 1 inner
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.d 4
13.b even 2 1 inner 507.2.j.d 4
13.c even 3 1 507.2.b.b 2
13.c even 3 1 inner 507.2.j.d 4
13.d odd 4 1 39.2.e.a 2
13.d odd 4 1 507.2.e.c 2
13.e even 6 1 507.2.b.b 2
13.e even 6 1 inner 507.2.j.d 4
13.f odd 12 1 39.2.e.a 2
13.f odd 12 1 507.2.a.b 1
13.f odd 12 1 507.2.a.c 1
13.f odd 12 1 507.2.e.c 2
39.f even 4 1 117.2.g.b 2
39.h odd 6 1 1521.2.b.c 2
39.i odd 6 1 1521.2.b.c 2
39.k even 12 1 117.2.g.b 2
39.k even 12 1 1521.2.a.a 1
39.k even 12 1 1521.2.a.d 1
52.f even 4 1 624.2.q.c 2
52.l even 12 1 624.2.q.c 2
52.l even 12 1 8112.2.a.w 1
52.l even 12 1 8112.2.a.bc 1
65.f even 4 1 975.2.bb.d 4
65.g odd 4 1 975.2.i.f 2
65.k even 4 1 975.2.bb.d 4
65.o even 12 1 975.2.bb.d 4
65.s odd 12 1 975.2.i.f 2
65.t even 12 1 975.2.bb.d 4
156.l odd 4 1 1872.2.t.j 2
156.v odd 12 1 1872.2.t.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 13.d odd 4 1
39.2.e.a 2 13.f odd 12 1
117.2.g.b 2 39.f even 4 1
117.2.g.b 2 39.k even 12 1
507.2.a.b 1 13.f odd 12 1
507.2.a.c 1 13.f odd 12 1
507.2.b.b 2 13.c even 3 1
507.2.b.b 2 13.e even 6 1
507.2.e.c 2 13.d odd 4 1
507.2.e.c 2 13.f odd 12 1
507.2.j.d 4 1.a even 1 1 trivial
507.2.j.d 4 13.b even 2 1 inner
507.2.j.d 4 13.c even 3 1 inner
507.2.j.d 4 13.e even 6 1 inner
624.2.q.c 2 52.f even 4 1
624.2.q.c 2 52.l even 12 1
975.2.i.f 2 65.g odd 4 1
975.2.i.f 2 65.s odd 12 1
975.2.bb.d 4 65.f even 4 1
975.2.bb.d 4 65.k even 4 1
975.2.bb.d 4 65.o even 12 1
975.2.bb.d 4 65.t even 12 1
1521.2.a.a 1 39.k even 12 1
1521.2.a.d 1 39.k even 12 1
1521.2.b.c 2 39.h odd 6 1
1521.2.b.c 2 39.i odd 6 1
1872.2.t.j 2 156.l odd 4 1
1872.2.t.j 2 156.v odd 12 1
8112.2.a.w 1 52.l even 12 1
8112.2.a.bc 1 52.l even 12 1

## Hecke kernels

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

 $$T_{2}^{4} - T_{2}^{2} + 1$$ T2^4 - T2^2 + 1 $$T_{5}^{2} + 1$$ T5^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - 4T^{2} + 16$$
$11$ $$T^{4} - 4T^{2} + 16$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 7 T + 49)^{2}$$
$19$ $$T^{4} - 36T^{2} + 1296$$
$23$ $$(T^{2} + 6 T + 36)^{2}$$
$29$ $$(T^{2} - T + 1)^{2}$$
$31$ $$(T^{2} + 16)^{2}$$
$37$ $$T^{4} - T^{2} + 1$$
$41$ $$T^{4} - 81T^{2} + 6561$$
$43$ $$(T^{2} - 6 T + 36)^{2}$$
$47$ $$(T^{2} + 36)^{2}$$
$53$ $$(T + 9)^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + T + 1)^{2}$$
$67$ $$T^{4} - 4T^{2} + 16$$
$71$ $$T^{4} - 36T^{2} + 1296$$
$73$ $$(T^{2} + 121)^{2}$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} + 196)^{2}$$
$89$ $$T^{4} - 196 T^{2} + 38416$$
$97$ $$T^{4} - 4T^{2} + 16$$