# Properties

 Label 462.2.j.c Level $462$ Weight $2$ Character orbit 462.j Analytic conductor $3.689$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.j (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + (\zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{5} + \zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} - \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10})$$ q + (-z^3 + z^2 - z + 1) * q^2 + z^2 * q^3 - z^3 * q^4 + (z^2 + 2*z + 1) * q^5 + z * q^6 + z^3 * q^7 - z^2 * q^8 + (z^3 - z^2 + z - 1) * q^9 $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + (\zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{5} + \zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} - \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 3) q^{10} + (\zeta_{10}^{3} + 3 \zeta_{10}^{2} - \zeta_{10} + 1) q^{11} + q^{12} + (2 \zeta_{10}^{3} - 2) q^{13} + \zeta_{10}^{2} q^{14} + (3 \zeta_{10}^{3} + \zeta_{10} - 1) q^{15} - \zeta_{10} q^{16} + (\zeta_{10}^{2} - 7 \zeta_{10} + 1) q^{17} + \zeta_{10}^{3} q^{18} + ( - 5 \zeta_{10}^{3} + \zeta_{10}^{2} - 5 \zeta_{10}) q^{19} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{20} - q^{21} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{22} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + 2) q^{23} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{24} + (5 \zeta_{10}^{3} + 5 \zeta_{10}) q^{25} + (2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{26} - \zeta_{10} q^{27} + \zeta_{10} q^{28} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{29} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} + \zeta_{10}) q^{30} + ( - 7 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 7) q^{31} - q^{32} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{33} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 6) q^{34} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{35} + \zeta_{10}^{2} q^{36} + (7 \zeta_{10}^{3} - \zeta_{10} + 1) q^{37} + ( - 5 \zeta_{10}^{2} + \zeta_{10} - 5) q^{38} + ( - 2 \zeta_{10}^{2} - 2) q^{39} + ( - 3 \zeta_{10}^{3} - \zeta_{10} + 1) q^{40} + ( - 3 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 3 \zeta_{10}) q^{41} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{42} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2) q^{43} + ( - \zeta_{10}^{2} + 2 \zeta_{10} + 2) q^{44} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 3) q^{45} + ( - 2 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} + 2) q^{46} + ( - 10 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 10 \zeta_{10}) q^{47} - \zeta_{10}^{3} q^{48} - \zeta_{10} q^{49} + (5 \zeta_{10}^{2} + 5) q^{50} + ( - 6 \zeta_{10}^{3} + \zeta_{10} - 1) q^{51} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{52} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{53} - q^{54} + (11 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 6 \zeta_{10} - 5) q^{55} + q^{56} + ( - 4 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 4) q^{57} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{58} + ( - 4 \zeta_{10}^{3} - 8 \zeta_{10} + 8) q^{59} + (\zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{60} + ( - 2 \zeta_{10}^{2} + 8 \zeta_{10} - 2) q^{61} + ( - 7 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{62} - \zeta_{10}^{2} q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 8) q^{65} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{66} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{67} + (6 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 6) q^{68} + (7 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 7 \zeta_{10}) q^{69} + (3 \zeta_{10}^{3} + \zeta_{10} - 1) q^{70} + ( - 4 \zeta_{10}^{2} - 4 \zeta_{10} - 4) q^{71} + \zeta_{10} q^{72} + 8 \zeta_{10}^{3} q^{73} + ( - \zeta_{10}^{3} + 8 \zeta_{10}^{2} - \zeta_{10}) q^{74} + (5 \zeta_{10}^{3} - 5) q^{75} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 4) q^{76} + (\zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{77} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{78} + (2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 2) q^{79} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10}) q^{80} - \zeta_{10}^{3} q^{81} + ( - 3 \zeta_{10}^{2} + 10 \zeta_{10} - 3) q^{82} + (2 \zeta_{10}^{2} - 14 \zeta_{10} + 2) q^{83} + \zeta_{10}^{3} q^{84} + ( - 4 \zeta_{10}^{3} - 13 \zeta_{10}^{2} - 4 \zeta_{10}) q^{85} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{86} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{87} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 3 \zeta_{10} + 4) q^{88} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 10) q^{89} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{90} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{91} + ( - 2 \zeta_{10}^{3} - 7 \zeta_{10} + 7) q^{92} + (5 \zeta_{10}^{2} + 2 \zeta_{10} + 5) q^{93} + ( - 10 \zeta_{10}^{2} + 4 \zeta_{10} - 10) q^{94} + ( - 17 \zeta_{10}^{3} - 14 \zeta_{10} + 14) q^{95} - \zeta_{10}^{2} q^{96} + (8 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 8) q^{97} - q^{98} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{99} +O(q^{100})$$ q + (-z^3 + z^2 - z + 1) * q^2 + z^2 * q^3 - z^3 * q^4 + (z^2 + 2*z + 1) * q^5 + z * q^6 + z^3 * q^7 - z^2 * q^8 + (z^3 - z^2 + z - 1) * q^9 + (-z^3 + z^2 + 3) * q^10 + (z^3 + 3*z^2 - z + 1) * q^11 + q^12 + (2*z^3 - 2) * q^13 + z^2 * q^14 + (3*z^3 + z - 1) * q^15 - z * q^16 + (z^2 - 7*z + 1) * q^17 + z^3 * q^18 + (-5*z^3 + z^2 - 5*z) * q^19 + (-3*z^3 + 2*z^2 - 2*z + 3) * q^20 - q^21 + (-z^3 + 2*z^2 + 2*z) * q^22 + (-7*z^3 + 7*z^2 + 2) * q^23 + (-z^3 + z^2 - z + 1) * q^24 + (5*z^3 + 5*z) * q^25 + (2*z^3 + 2*z - 2) * q^26 - z * q^27 + z * q^28 + (-2*z^3 - 2*z + 2) * q^29 + (z^3 + 2*z^2 + z) * q^30 + (-7*z^3 + 2*z^2 - 2*z + 7) * q^31 - q^32 + (2*z^3 - 2*z^2 + 3*z - 4) * q^33 + (-z^3 + z^2 - 6) * q^34 + (3*z^3 - 2*z^2 + 2*z - 3) * q^35 + z^2 * q^36 + (7*z^3 - z + 1) * q^37 + (-5*z^2 + z - 5) * q^38 + (-2*z^2 - 2) * q^39 + (-3*z^3 - z + 1) * q^40 + (-3*z^3 + 10*z^2 - 3*z) * q^41 + (z^3 - z^2 + z - 1) * q^42 + (2*z^3 - 2*z^2 + 2) * q^43 + (-z^2 + 2*z + 2) * q^44 + (z^3 - z^2 - 3) * q^45 + (-2*z^3 - 5*z^2 + 5*z + 2) * q^46 + (-10*z^3 + 4*z^2 - 10*z) * q^47 - z^3 * q^48 - z * q^49 + (5*z^2 + 5) * q^50 + (-6*z^3 + z - 1) * q^51 + (2*z^3 + 2*z) * q^52 + (-4*z^3 - 4*z^2 + 4*z + 4) * q^53 - q^54 + (11*z^3 - 3*z^2 + 6*z - 5) * q^55 + q^56 + (-4*z^3 - z^2 + z + 4) * q^57 + (-2*z^3 - 2*z) * q^58 + (-4*z^3 - 8*z + 8) * q^59 + (z^2 + 2*z + 1) * q^60 + (-2*z^2 + 8*z - 2) * q^61 + (-7*z^3 - 5*z + 5) * q^62 - z^2 * q^63 + (z^3 - z^2 + z - 1) * q^64 + (6*z^3 - 6*z^2 - 8) * q^65 + (4*z^3 - 2*z^2 + 2*z - 1) * q^66 + (-2*z^3 + 2*z^2 + 2) * q^67 + (6*z^3 - 7*z^2 + 7*z - 6) * q^68 + (7*z^3 - 5*z^2 + 7*z) * q^69 + (3*z^3 + z - 1) * q^70 + (-4*z^2 - 4*z - 4) * q^71 + z * q^72 + 8*z^3 * q^73 + (-z^3 + 8*z^2 - z) * q^74 + (5*z^3 - 5) * q^75 + (5*z^3 - 5*z^2 - 4) * q^76 + (z^2 - 2*z - 2) * q^77 + (2*z^3 - 2*z^2 - 2) * q^78 + (2*z^3 + 4*z^2 - 4*z - 2) * q^79 + (-z^3 - 2*z^2 - z) * q^80 - z^3 * q^81 + (-3*z^2 + 10*z - 3) * q^82 + (2*z^2 - 14*z + 2) * q^83 + z^3 * q^84 + (-4*z^3 - 13*z^2 - 4*z) * q^85 + (-2*z^3 + 4*z^2 - 4*z + 2) * q^86 + (-2*z^3 + 2*z^2 + 2) * q^87 + (-2*z^3 + 2*z^2 - 3*z + 4) * q^88 + (3*z^3 - 3*z^2 - 10) * q^89 + (3*z^3 - 2*z^2 + 2*z - 3) * q^90 + (-2*z^3 - 2*z) * q^91 + (-2*z^3 - 7*z + 7) * q^92 + (5*z^2 + 2*z + 5) * q^93 + (-10*z^2 + 4*z - 10) * q^94 + (-17*z^3 - 14*z + 14) * q^95 - z^2 * q^96 + (8*z^3 - 10*z^2 + 10*z - 8) * q^97 - q^98 + (z^3 - 2*z^2 - 2*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - q^{3} - q^{4} + 5 q^{5} + q^{6} + q^{7} + q^{8} - q^{9}+O(q^{10})$$ 4 * q + q^2 - q^3 - q^4 + 5 * q^5 + q^6 + q^7 + q^8 - q^9 $$4 q + q^{2} - q^{3} - q^{4} + 5 q^{5} + q^{6} + q^{7} + q^{8} - q^{9} + 10 q^{10} + q^{11} + 4 q^{12} - 6 q^{13} - q^{14} - q^{16} - 4 q^{17} + q^{18} - 11 q^{19} + 5 q^{20} - 4 q^{21} - q^{22} - 6 q^{23} + q^{24} + 10 q^{25} - 4 q^{26} - q^{27} + q^{28} + 4 q^{29} + 17 q^{31} - 4 q^{32} - 9 q^{33} - 26 q^{34} - 5 q^{35} - q^{36} + 10 q^{37} - 14 q^{38} - 6 q^{39} - 16 q^{41} - q^{42} + 12 q^{43} + 11 q^{44} - 10 q^{45} + 16 q^{46} - 24 q^{47} - q^{48} - q^{49} + 15 q^{50} - 9 q^{51} + 4 q^{52} + 20 q^{53} - 4 q^{54} + 4 q^{56} + 14 q^{57} - 4 q^{58} + 20 q^{59} + 5 q^{60} + 2 q^{61} + 8 q^{62} + q^{63} - q^{64} - 20 q^{65} + 4 q^{66} + 4 q^{67} - 4 q^{68} + 19 q^{69} - 16 q^{71} + q^{72} + 8 q^{73} - 10 q^{74} - 15 q^{75} - 6 q^{76} - 11 q^{77} - 4 q^{78} - 14 q^{79} - q^{81} + q^{82} - 8 q^{83} + q^{84} + 5 q^{85} - 2 q^{86} + 4 q^{87} + 9 q^{88} - 34 q^{89} - 5 q^{90} - 4 q^{91} + 19 q^{92} + 17 q^{93} - 26 q^{94} + 25 q^{95} + q^{96} - 4 q^{97} - 4 q^{98} + q^{99}+O(q^{100})$$ 4 * q + q^2 - q^3 - q^4 + 5 * q^5 + q^6 + q^7 + q^8 - q^9 + 10 * q^10 + q^11 + 4 * q^12 - 6 * q^13 - q^14 - q^16 - 4 * q^17 + q^18 - 11 * q^19 + 5 * q^20 - 4 * q^21 - q^22 - 6 * q^23 + q^24 + 10 * q^25 - 4 * q^26 - q^27 + q^28 + 4 * q^29 + 17 * q^31 - 4 * q^32 - 9 * q^33 - 26 * q^34 - 5 * q^35 - q^36 + 10 * q^37 - 14 * q^38 - 6 * q^39 - 16 * q^41 - q^42 + 12 * q^43 + 11 * q^44 - 10 * q^45 + 16 * q^46 - 24 * q^47 - q^48 - q^49 + 15 * q^50 - 9 * q^51 + 4 * q^52 + 20 * q^53 - 4 * q^54 + 4 * q^56 + 14 * q^57 - 4 * q^58 + 20 * q^59 + 5 * q^60 + 2 * q^61 + 8 * q^62 + q^63 - q^64 - 20 * q^65 + 4 * q^66 + 4 * q^67 - 4 * q^68 + 19 * q^69 - 16 * q^71 + q^72 + 8 * q^73 - 10 * q^74 - 15 * q^75 - 6 * q^76 - 11 * q^77 - 4 * q^78 - 14 * q^79 - q^81 + q^82 - 8 * q^83 + q^84 + 5 * q^85 - 2 * q^86 + 4 * q^87 + 9 * q^88 - 34 * q^89 - 5 * q^90 - 4 * q^91 + 19 * q^92 + 17 * q^93 - 26 * q^94 + 25 * q^95 + q^96 - 4 * q^97 - 4 * q^98 + q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i
0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 2.92705 2.12663i 0.809017 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i 3.61803
295.1 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i −0.427051 1.31433i −0.309017 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 0.951057i 1.38197
379.1 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i −0.427051 + 1.31433i −0.309017 + 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i 1.38197
421.1 0.809017 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 2.92705 + 2.12663i 0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i 3.61803
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.j.c 4
11.c even 5 1 inner 462.2.j.c 4
11.c even 5 1 5082.2.a.bh 2
11.d odd 10 1 5082.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.c 4 1.a even 1 1 trivial
462.2.j.c 4 11.c even 5 1 inner
5082.2.a.bh 2 11.c even 5 1
5082.2.a.br 2 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 5T_{5}^{3} + 10T_{5}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$3$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$5$ $$T^{4} - 5 T^{3} + 10 T^{2} + 25$$
$7$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$11$ $$T^{4} - T^{3} + 21 T^{2} - 11 T + 121$$
$13$ $$T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16$$
$17$ $$T^{4} + 4 T^{3} + 46 T^{2} + \cdots + 1681$$
$19$ $$T^{4} + 11 T^{3} + 96 T^{2} + \cdots + 841$$
$23$ $$(T^{2} + 3 T - 59)^{2}$$
$29$ $$T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16$$
$31$ $$T^{4} - 17 T^{3} + 114 T^{2} + \cdots + 121$$
$37$ $$T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 3025$$
$41$ $$T^{4} + 16 T^{3} + 106 T^{2} + \cdots + 3721$$
$43$ $$(T^{2} - 6 T + 4)^{2}$$
$47$ $$T^{4} + 24 T^{3} + 376 T^{2} + \cdots + 15376$$
$53$ $$T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400$$
$59$ $$T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400$$
$61$ $$T^{4} - 2 T^{3} + 64 T^{2} + \cdots + 1936$$
$67$ $$(T^{2} - 2 T - 4)^{2}$$
$71$ $$T^{4} + 16 T^{3} + 96 T^{2} + \cdots + 256$$
$73$ $$T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$79$ $$T^{4} + 14 T^{3} + 136 T^{2} + \cdots + 1936$$
$83$ $$T^{4} + 8 T^{3} + 184 T^{2} + \cdots + 26896$$
$89$ $$(T^{2} + 17 T + 61)^{2}$$
$97$ $$T^{4} + 4 T^{3} + 96 T^{2} + \cdots + 5776$$