# Properties

 Label 507.2.e.h Level $507$ Weight $2$ Character orbit 507.e Analytic conductor $4.048$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + ( - \beta_{2} - 1) q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} - 2 \beta_{3} q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{6} + ( - 2 \beta_{3} - 2 \beta_1) q^{7} + (\beta_{3} - 3) q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + (-b2 - 1) * q^3 + (2*b3 + b2 + 2*b1) * q^4 - 2*b3 * q^5 + (-b3 - b2 - b1) * q^6 + (-2*b3 - 2*b1) * q^7 + (b3 - 3) * q^8 + b2 * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + ( - \beta_{2} - 1) q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} - 2 \beta_{3} q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{6} + ( - 2 \beta_{3} - 2 \beta_1) q^{7} + (\beta_{3} - 3) q^{8} + \beta_{2} q^{9} + (4 \beta_{2} + 2 \beta_1 + 4) q^{10} + (2 \beta_{2} + 2) q^{11} + ( - 2 \beta_{3} + 1) q^{12} + ( - 2 \beta_{3} + 4) q^{14} - 2 \beta_1 q^{15} + ( - 3 \beta_{2} - 3) q^{16} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{17} + (\beta_{3} - 1) q^{18} + (2 \beta_{3} + 2 \beta_1) q^{19} + (2 \beta_{3} + 8 \beta_{2} + 2 \beta_1) q^{20} + 2 \beta_{3} q^{21} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{22} + (4 \beta_{2} + 4) q^{23} + (3 \beta_{2} + \beta_1 + 3) q^{24} + 3 q^{25} + q^{27} + (8 \beta_{2} + 2 \beta_1 + 8) q^{28} + ( - 2 \beta_{2} - 2) q^{29} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{30} + (2 \beta_{3} - 4) q^{31} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{32} - 2 \beta_{2} q^{33} + ( - 2 \beta_{3} + 6) q^{34} - 8 \beta_{2} q^{35} + ( - \beta_{2} - 2 \beta_1 - 1) q^{36} + (2 \beta_{2} - 4 \beta_1 + 2) q^{37} + (2 \beta_{3} - 4) q^{38} + (6 \beta_{3} - 4) q^{40} + ( - 8 \beta_{2} - 2 \beta_1 - 8) q^{41} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{42} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{43} + (4 \beta_{3} - 2) q^{44} + (2 \beta_{3} + 2 \beta_1) q^{45} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{46} + ( - 4 \beta_{3} - 6) q^{47} + 3 \beta_{2} q^{48} + ( - \beta_{2} - 1) q^{49} + (3 \beta_{2} + 3 \beta_1 + 3) q^{50} + (4 \beta_{3} + 2) q^{51} - 2 q^{53} + (\beta_{2} + \beta_1 + 1) q^{54} + 4 \beta_1 q^{55} + (6 \beta_{3} + 4 \beta_{2} + 6 \beta_1) q^{56} - 2 \beta_{3} q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{58} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{59} + ( - 2 \beta_{3} + 8) q^{60} + ( - 8 \beta_{3} + 2 \beta_{2} - 8 \beta_1) q^{61} + ( - 8 \beta_{2} - 6 \beta_1 - 8) q^{62} + 2 \beta_1 q^{63} + (2 \beta_{3} - 7) q^{64} + ( - 2 \beta_{3} + 2) q^{66} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{67} + (14 \beta_{2} + 14) q^{68} - 4 \beta_{2} q^{69} + ( - 8 \beta_{3} + 8) q^{70} + 2 \beta_{2} q^{71} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{72} + ( - 4 \beta_{3} + 6) q^{73} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{74} + ( - 3 \beta_{2} - 3) q^{75} + ( - 8 \beta_{2} - 2 \beta_1 - 8) q^{76} - 4 \beta_{3} q^{77} - 8 \beta_{3} q^{79} - 6 \beta_1 q^{80} + ( - \beta_{2} - 1) q^{81} + ( - 10 \beta_{3} - 12 \beta_{2} - 10 \beta_1) q^{82} + (4 \beta_{3} - 2) q^{83} + ( - 2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{84} + (4 \beta_{3} - 16 \beta_{2} + 4 \beta_1) q^{85} + (8 \beta_{3} - 12) q^{86} + 2 \beta_{2} q^{87} + ( - 6 \beta_{2} - 2 \beta_1 - 6) q^{88} + ( - 12 \beta_{2} + 2 \beta_1 - 12) q^{89} + (2 \beta_{3} - 4) q^{90} + (8 \beta_{3} - 4) q^{92} + (4 \beta_{2} + 2 \beta_1 + 4) q^{93} + (2 \beta_{2} - 2 \beta_1 + 2) q^{94} + 8 \beta_{2} q^{95} + (\beta_{3} + 3) q^{96} + ( - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{97} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{98} - 2 q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + (-b2 - 1) * q^3 + (2*b3 + b2 + 2*b1) * q^4 - 2*b3 * q^5 + (-b3 - b2 - b1) * q^6 + (-2*b3 - 2*b1) * q^7 + (b3 - 3) * q^8 + b2 * q^9 + (4*b2 + 2*b1 + 4) * q^10 + (2*b2 + 2) * q^11 + (-2*b3 + 1) * q^12 + (-2*b3 + 4) * q^14 - 2*b1 * q^15 + (-3*b2 - 3) * q^16 + (-4*b3 + 2*b2 - 4*b1) * q^17 + (b3 - 1) * q^18 + (2*b3 + 2*b1) * q^19 + (2*b3 + 8*b2 + 2*b1) * q^20 + 2*b3 * q^21 + (2*b3 + 2*b2 + 2*b1) * q^22 + (4*b2 + 4) * q^23 + (3*b2 + b1 + 3) * q^24 + 3 * q^25 + q^27 + (8*b2 + 2*b1 + 8) * q^28 + (-2*b2 - 2) * q^29 + (-2*b3 - 4*b2 - 2*b1) * q^30 + (2*b3 - 4) * q^31 + (-b3 + 3*b2 - b1) * q^32 - 2*b2 * q^33 + (-2*b3 + 6) * q^34 - 8*b2 * q^35 + (-b2 - 2*b1 - 1) * q^36 + (2*b2 - 4*b1 + 2) * q^37 + (2*b3 - 4) * q^38 + (6*b3 - 4) * q^40 + (-8*b2 - 2*b1 - 8) * q^41 + (-4*b2 - 2*b1 - 4) * q^42 + (4*b3 + 4*b2 + 4*b1) * q^43 + (4*b3 - 2) * q^44 + (2*b3 + 2*b1) * q^45 + (4*b3 + 4*b2 + 4*b1) * q^46 + (-4*b3 - 6) * q^47 + 3*b2 * q^48 + (-b2 - 1) * q^49 + (3*b2 + 3*b1 + 3) * q^50 + (4*b3 + 2) * q^51 - 2 * q^53 + (b2 + b1 + 1) * q^54 + 4*b1 * q^55 + (6*b3 + 4*b2 + 6*b1) * q^56 - 2*b3 * q^57 + (-2*b3 - 2*b2 - 2*b1) * q^58 + (-4*b3 + 2*b2 - 4*b1) * q^59 + (-2*b3 + 8) * q^60 + (-8*b3 + 2*b2 - 8*b1) * q^61 + (-8*b2 - 6*b1 - 8) * q^62 + 2*b1 * q^63 + (2*b3 - 7) * q^64 + (-2*b3 + 2) * q^66 + (-4*b2 + 2*b1 - 4) * q^67 + (14*b2 + 14) * q^68 - 4*b2 * q^69 + (-8*b3 + 8) * q^70 + 2*b2 * q^71 + (-b3 - 3*b2 - b1) * q^72 + (-4*b3 + 6) * q^73 + (-2*b3 - 6*b2 - 2*b1) * q^74 + (-3*b2 - 3) * q^75 + (-8*b2 - 2*b1 - 8) * q^76 - 4*b3 * q^77 - 8*b3 * q^79 - 6*b1 * q^80 + (-b2 - 1) * q^81 + (-10*b3 - 12*b2 - 10*b1) * q^82 + (4*b3 - 2) * q^83 + (-2*b3 - 8*b2 - 2*b1) * q^84 + (4*b3 - 16*b2 + 4*b1) * q^85 + (8*b3 - 12) * q^86 + 2*b2 * q^87 + (-6*b2 - 2*b1 - 6) * q^88 + (-12*b2 + 2*b1 - 12) * q^89 + (2*b3 - 4) * q^90 + (8*b3 - 4) * q^92 + (4*b2 + 2*b1 + 4) * q^93 + (2*b2 - 2*b1 + 2) * q^94 + 8*b2 * q^95 + (b3 + 3) * q^96 + (-4*b3 - 2*b2 - 4*b1) * q^97 + (-b3 - b2 - b1) * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} - 12 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 - 12 * q^8 - 2 * q^9 $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} - 12 q^{8} - 2 q^{9} + 8 q^{10} + 4 q^{11} + 4 q^{12} + 16 q^{14} - 6 q^{16} - 4 q^{17} - 4 q^{18} - 16 q^{20} - 4 q^{22} + 8 q^{23} + 6 q^{24} + 12 q^{25} + 4 q^{27} + 16 q^{28} - 4 q^{29} + 8 q^{30} - 16 q^{31} - 6 q^{32} + 4 q^{33} + 24 q^{34} + 16 q^{35} - 2 q^{36} + 4 q^{37} - 16 q^{38} - 16 q^{40} - 16 q^{41} - 8 q^{42} - 8 q^{43} - 8 q^{44} - 8 q^{46} - 24 q^{47} - 6 q^{48} - 2 q^{49} + 6 q^{50} + 8 q^{51} - 8 q^{53} + 2 q^{54} - 8 q^{56} + 4 q^{58} - 4 q^{59} + 32 q^{60} - 4 q^{61} - 16 q^{62} - 28 q^{64} + 8 q^{66} - 8 q^{67} + 28 q^{68} + 8 q^{69} + 32 q^{70} - 4 q^{71} + 6 q^{72} + 24 q^{73} + 12 q^{74} - 6 q^{75} - 16 q^{76} - 2 q^{81} + 24 q^{82} - 8 q^{83} + 16 q^{84} + 32 q^{85} - 48 q^{86} - 4 q^{87} - 12 q^{88} - 24 q^{89} - 16 q^{90} - 16 q^{92} + 8 q^{93} + 4 q^{94} - 16 q^{95} + 12 q^{96} + 4 q^{97} + 2 q^{98} - 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 - 12 * q^8 - 2 * q^9 + 8 * q^10 + 4 * q^11 + 4 * q^12 + 16 * q^14 - 6 * q^16 - 4 * q^17 - 4 * q^18 - 16 * q^20 - 4 * q^22 + 8 * q^23 + 6 * q^24 + 12 * q^25 + 4 * q^27 + 16 * q^28 - 4 * q^29 + 8 * q^30 - 16 * q^31 - 6 * q^32 + 4 * q^33 + 24 * q^34 + 16 * q^35 - 2 * q^36 + 4 * q^37 - 16 * q^38 - 16 * q^40 - 16 * q^41 - 8 * q^42 - 8 * q^43 - 8 * q^44 - 8 * q^46 - 24 * q^47 - 6 * q^48 - 2 * q^49 + 6 * q^50 + 8 * q^51 - 8 * q^53 + 2 * q^54 - 8 * q^56 + 4 * q^58 - 4 * q^59 + 32 * q^60 - 4 * q^61 - 16 * q^62 - 28 * q^64 + 8 * q^66 - 8 * q^67 + 28 * q^68 + 8 * q^69 + 32 * q^70 - 4 * q^71 + 6 * q^72 + 24 * q^73 + 12 * q^74 - 6 * q^75 - 16 * q^76 - 2 * q^81 + 24 * q^82 - 8 * q^83 + 16 * q^84 + 32 * q^85 - 48 * q^86 - 4 * q^87 - 12 * q^88 - 24 * q^89 - 16 * q^90 - 16 * q^92 + 8 * q^93 + 4 * q^94 - 16 * q^95 + 12 * q^96 + 4 * q^97 + 2 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.207107 + 0.358719i −0.500000 + 0.866025i 0.914214 + 1.58346i −2.82843 −0.207107 0.358719i −1.41421 2.44949i −1.58579 −0.500000 0.866025i 0.585786 1.01461i
22.2 1.20711 2.09077i −0.500000 + 0.866025i −1.91421 3.31552i 2.82843 1.20711 + 2.09077i 1.41421 + 2.44949i −4.41421 −0.500000 0.866025i 3.41421 5.91359i
484.1 −0.207107 0.358719i −0.500000 0.866025i 0.914214 1.58346i −2.82843 −0.207107 + 0.358719i −1.41421 + 2.44949i −1.58579 −0.500000 + 0.866025i 0.585786 + 1.01461i
484.2 1.20711 + 2.09077i −0.500000 0.866025i −1.91421 + 3.31552i 2.82843 1.20711 2.09077i 1.41421 2.44949i −4.41421 −0.500000 + 0.866025i 3.41421 + 5.91359i
 $$n$$: 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.h 4
13.b even 2 1 507.2.e.d 4
13.c even 3 1 39.2.a.b 2
13.c even 3 1 inner 507.2.e.h 4
13.d odd 4 2 507.2.j.f 8
13.e even 6 1 507.2.a.h 2
13.e even 6 1 507.2.e.d 4
13.f odd 12 2 507.2.b.e 4
13.f odd 12 2 507.2.j.f 8
39.h odd 6 1 1521.2.a.f 2
39.i odd 6 1 117.2.a.c 2
39.k even 12 2 1521.2.b.j 4
52.i odd 6 1 8112.2.a.bm 2
52.j odd 6 1 624.2.a.k 2
65.n even 6 1 975.2.a.l 2
65.q odd 12 2 975.2.c.h 4
91.n odd 6 1 1911.2.a.h 2
104.n odd 6 1 2496.2.a.bi 2
104.r even 6 1 2496.2.a.bf 2
117.f even 3 1 1053.2.e.m 4
117.h even 3 1 1053.2.e.m 4
117.k odd 6 1 1053.2.e.e 4
117.u odd 6 1 1053.2.e.e 4
143.k odd 6 1 4719.2.a.p 2
156.p even 6 1 1872.2.a.w 2
195.x odd 6 1 2925.2.a.v 2
195.bl even 12 2 2925.2.c.u 4
273.bn even 6 1 5733.2.a.u 2
312.bh odd 6 1 7488.2.a.cl 2
312.bn even 6 1 7488.2.a.co 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 13.c even 3 1
117.2.a.c 2 39.i odd 6 1
507.2.a.h 2 13.e even 6 1
507.2.b.e 4 13.f odd 12 2
507.2.e.d 4 13.b even 2 1
507.2.e.d 4 13.e even 6 1
507.2.e.h 4 1.a even 1 1 trivial
507.2.e.h 4 13.c even 3 1 inner
507.2.j.f 8 13.d odd 4 2
507.2.j.f 8 13.f odd 12 2
624.2.a.k 2 52.j odd 6 1
975.2.a.l 2 65.n even 6 1
975.2.c.h 4 65.q odd 12 2
1053.2.e.e 4 117.k odd 6 1
1053.2.e.e 4 117.u odd 6 1
1053.2.e.m 4 117.f even 3 1
1053.2.e.m 4 117.h even 3 1
1521.2.a.f 2 39.h odd 6 1
1521.2.b.j 4 39.k even 12 2
1872.2.a.w 2 156.p even 6 1
1911.2.a.h 2 91.n odd 6 1
2496.2.a.bf 2 104.r even 6 1
2496.2.a.bi 2 104.n odd 6 1
2925.2.a.v 2 195.x odd 6 1
2925.2.c.u 4 195.bl even 12 2
4719.2.a.p 2 143.k odd 6 1
5733.2.a.u 2 273.bn even 6 1
7488.2.a.cl 2 312.bh odd 6 1
7488.2.a.co 2 312.bn even 6 1
8112.2.a.bm 2 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1$$ T2^4 - 2*T2^3 + 5*T2^2 + 2*T2 + 1 $$T_{5}^{2} - 8$$ T5^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} - 8)^{2}$$
$7$ $$T^{4} + 8T^{2} + 64$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 4 T^{3} + 44 T^{2} - 112 T + 784$$
$19$ $$T^{4} + 8T^{2} + 64$$
$23$ $$(T^{2} - 4 T + 16)^{2}$$
$29$ $$(T^{2} + 2 T + 4)^{2}$$
$31$ $$(T^{2} + 8 T + 8)^{2}$$
$37$ $$T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784$$
$41$ $$T^{4} + 16 T^{3} + 200 T^{2} + \cdots + 3136$$
$43$ $$T^{4} + 8 T^{3} + 80 T^{2} - 128 T + 256$$
$47$ $$(T^{2} + 12 T + 4)^{2}$$
$53$ $$(T + 2)^{4}$$
$59$ $$T^{4} + 4 T^{3} + 44 T^{2} - 112 T + 784$$
$61$ $$T^{4} + 4 T^{3} + 140 T^{2} + \cdots + 15376$$
$67$ $$T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64$$
$71$ $$(T^{2} + 2 T + 4)^{2}$$
$73$ $$(T^{2} - 12 T + 4)^{2}$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$(T^{2} + 4 T - 28)^{2}$$
$89$ $$T^{4} + 24 T^{3} + 440 T^{2} + \cdots + 18496$$
$97$ $$T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784$$