# Properties

 Label 462.2.j.d 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + ( - \zeta_{10}^{2} - 1) q^{5} - \zeta_{10} q^{6} - \zeta_{10}^{3} q^{7} - \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10})$$ q + (-z^3 + z^2 - z + 1) * q^2 - z^2 * q^3 - z^3 * q^4 + (-z^2 - 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} - 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} - 1) q^{10} + ( - \zeta_{10}^{3} - 3 \zeta_{10}^{2} + \zeta_{10} - 1) q^{11} - q^{12} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{13} - \zeta_{10}^{2} q^{14} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{15} - \zeta_{10} q^{16} + ( - \zeta_{10}^{2} - \zeta_{10} - 1) q^{17} + \zeta_{10}^{3} q^{18} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10}) q^{19} + (\zeta_{10}^{3} - 1) q^{20} - q^{21} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{22} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2}) q^{23} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{24} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10}) q^{25} + 2 \zeta_{10}^{3} q^{26} + \zeta_{10} q^{27} - \zeta_{10} q^{28} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{29} + (\zeta_{10}^{3} + \zeta_{10}) q^{30} + (3 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 3) q^{31} - q^{32} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{33} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 2) q^{34} + (\zeta_{10}^{3} - 1) q^{35} + \zeta_{10}^{2} q^{36} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{37} + (3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{38} + 2 \zeta_{10} q^{39} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{40} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10}) q^{41} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{42} + 10 q^{43} + (\zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{44} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{45} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10}) q^{46} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{47} + \zeta_{10}^{3} q^{48} - \zeta_{10} q^{49} + (\zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{50} + (2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{51} + 2 \zeta_{10}^{2} q^{52} + ( - 6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 6) q^{53} + q^{54} + (3 \zeta_{10}^{3} + \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{55} - q^{56} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10}) q^{57} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}) q^{58} + (2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{59} + (\zeta_{10}^{2} + 1) q^{60} + (10 \zeta_{10}^{2} - 6 \zeta_{10} + 10) q^{61} + (3 \zeta_{10}^{3} - \zeta_{10} + 1) q^{62} + \zeta_{10}^{2} q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{65} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{66} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 4) q^{67} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{68} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{69} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{70} + (4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{71} + \zeta_{10} q^{72} + ( - 6 \zeta_{10}^{3} - 10 \zeta_{10} + 10) q^{73} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10}) q^{74} + (3 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 3) q^{75} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2}) q^{76} + (\zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{77} + 2 q^{78} + (8 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 8) q^{79} + (\zeta_{10}^{3} + \zeta_{10}) q^{80} - \zeta_{10}^{3} q^{81} + ( - \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{82} + ( - 2 \zeta_{10}^{2} - 4 \zeta_{10} - 2) q^{83} + \zeta_{10}^{3} q^{84} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} + 2 \zeta_{10}) q^{85} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 10) q^{86} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 4) q^{87} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{88} + ( - 11 \zeta_{10}^{3} + 11 \zeta_{10}^{2} + 2) q^{89} + ( - \zeta_{10}^{3} + 1) q^{90} + 2 \zeta_{10}^{2} q^{91} + ( - 3 \zeta_{10} + 3) q^{92} + ( - \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{93} + ( - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{94} - 3 \zeta_{10}^{3} q^{95} + \zeta_{10}^{2} q^{96} + (6 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 6) 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 - 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 - 1) * q^10 + (-z^3 - 3*z^2 + z - 1) * q^11 - q^12 + (2*z^3 - 2*z^2 + 2*z - 2) * q^13 - z^2 * q^14 + (z^3 + z - 1) * q^15 - z * q^16 + (-z^2 - z - 1) * q^17 + z^3 * q^18 + (3*z^3 - 3*z^2 + 3*z) * q^19 + (z^3 - 1) * q^20 - q^21 + (z^3 - 2*z^2 - 2*z) * q^22 + (-3*z^3 + 3*z^2) * q^23 + (z^3 - z^2 + z - 1) * q^24 + (z^3 - 4*z^2 + z) * q^25 + 2*z^3 * q^26 + z * q^27 - z * q^28 + (-4*z^3 - 4*z + 4) * q^29 + (z^3 + z) * q^30 + (3*z^3 - 4*z^2 + 4*z - 3) * q^31 - q^32 + (2*z^3 - 2*z^2 + 3*z - 4) * q^33 + (z^3 - z^2 - 2) * q^34 + (z^3 - 1) * q^35 + z^2 * q^36 + (z^3 + 3*z - 3) * q^37 + (3*z^2 - 3*z + 3) * q^38 + 2*z * q^39 + (z^3 + z - 1) * q^40 + (-z^3 + 2*z^2 - z) * q^41 + (z^3 - z^2 + z - 1) * q^42 + 10 * q^43 + (z^2 - 2*z - 2) * q^44 + (-z^3 + z^2 + 1) * q^45 + (-3*z^2 + 3*z) * q^46 + (-4*z^3 + 4*z^2 - 4*z) * q^47 + z^3 * q^48 - z * q^49 + (z^2 - 4*z + 1) * q^50 + (2*z^3 + z - 1) * q^51 + 2*z^2 * q^52 + (-6*z^3 + 2*z^2 - 2*z + 6) * q^53 + q^54 + (3*z^3 + z^2 + 2*z - 3) * q^55 - q^56 + (-3*z^2 + 3*z) * q^57 + (-4*z^3 - 4*z) * q^58 + (2*z^3 - 2*z + 2) * q^59 + (z^2 + 1) * q^60 + (10*z^2 - 6*z + 10) * q^61 + (3*z^3 - z + 1) * q^62 + z^2 * q^63 + (z^3 - z^2 + z - 1) * q^64 + (-2*z^3 + 2*z^2 + 2) * q^65 + (4*z^3 - 2*z^2 + 2*z - 1) * q^66 + (-10*z^3 + 10*z^2 + 4) * q^67 + (2*z^3 - z^2 + z - 2) * q^68 + (-3*z^3 + 3*z^2 - 3*z) * q^69 + (z^3 + z - 1) * q^70 + (4*z^2 - 4*z + 4) * q^71 + z * q^72 + (-6*z^3 - 10*z + 10) * q^73 + (3*z^3 - 2*z^2 + 3*z) * q^74 + (3*z^3 - 4*z^2 + 4*z - 3) * q^75 + (-3*z^3 + 3*z^2) * q^76 + (z^2 - 2*z - 2) * q^77 + 2 * q^78 + (8*z^3 - 12*z^2 + 12*z - 8) * q^79 + (z^3 + z) * q^80 - z^3 * q^81 + (-z^2 + 2*z - 1) * q^82 + (-2*z^2 - 4*z - 2) * q^83 + z^3 * q^84 + (2*z^3 + z^2 + 2*z) * q^85 + (-10*z^3 + 10*z^2 - 10*z + 10) * q^86 + (4*z^3 - 4*z^2 - 4) * q^87 + (2*z^3 - 2*z^2 + 3*z - 4) * q^88 + (-11*z^3 + 11*z^2 + 2) * q^89 + (-z^3 + 1) * q^90 + 2*z^2 * q^91 + (-3*z + 3) * q^92 + (-z^2 + 4*z - 1) * q^93 + (-4*z^2 + 4*z - 4) * q^94 - 3*z^3 * q^95 + z^2 * q^96 + (6*z^3 + 4*z^2 - 4*z - 6) * q^97 - q^98 + (-z^3 + 2*z^2 + 2*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{3} - q^{4} - 3 q^{5} - q^{6} - q^{7} + q^{8} - q^{9}+O(q^{10})$$ 4 * q + q^2 + q^3 - q^4 - 3 * q^5 - q^6 - q^7 + q^8 - q^9 $$4 q + q^{2} + q^{3} - q^{4} - 3 q^{5} - q^{6} - q^{7} + q^{8} - q^{9} - 2 q^{10} - q^{11} - 4 q^{12} - 2 q^{13} + q^{14} - 2 q^{15} - q^{16} - 4 q^{17} + q^{18} + 9 q^{19} - 3 q^{20} - 4 q^{21} + q^{22} - 6 q^{23} - q^{24} + 6 q^{25} + 2 q^{26} + q^{27} - q^{28} + 8 q^{29} + 2 q^{30} - q^{31} - 4 q^{32} - 9 q^{33} - 6 q^{34} - 3 q^{35} - q^{36} - 8 q^{37} + 6 q^{38} + 2 q^{39} - 2 q^{40} - 4 q^{41} - q^{42} + 40 q^{43} - 11 q^{44} + 2 q^{45} + 6 q^{46} - 12 q^{47} + q^{48} - q^{49} - q^{50} - q^{51} - 2 q^{52} + 14 q^{53} + 4 q^{54} - 8 q^{55} - 4 q^{56} + 6 q^{57} - 8 q^{58} + 8 q^{59} + 3 q^{60} + 24 q^{61} + 6 q^{62} - q^{63} - q^{64} + 4 q^{65} + 4 q^{66} - 4 q^{67} - 4 q^{68} - 9 q^{69} - 2 q^{70} + 8 q^{71} + q^{72} + 24 q^{73} + 8 q^{74} - q^{75} - 6 q^{76} - 11 q^{77} + 8 q^{78} + 2 q^{80} - q^{81} - q^{82} - 10 q^{83} + q^{84} + 3 q^{85} + 10 q^{86} - 8 q^{87} - 9 q^{88} - 14 q^{89} + 3 q^{90} - 2 q^{91} + 9 q^{92} + q^{93} - 8 q^{94} - 3 q^{95} - q^{96} - 26 q^{97} - 4 q^{98} - q^{99}+O(q^{100})$$ 4 * q + q^2 + q^3 - q^4 - 3 * q^5 - q^6 - q^7 + q^8 - q^9 - 2 * q^10 - q^11 - 4 * q^12 - 2 * q^13 + q^14 - 2 * q^15 - q^16 - 4 * q^17 + q^18 + 9 * q^19 - 3 * q^20 - 4 * q^21 + q^22 - 6 * q^23 - q^24 + 6 * q^25 + 2 * q^26 + q^27 - q^28 + 8 * q^29 + 2 * q^30 - q^31 - 4 * q^32 - 9 * q^33 - 6 * q^34 - 3 * q^35 - q^36 - 8 * q^37 + 6 * q^38 + 2 * q^39 - 2 * q^40 - 4 * q^41 - q^42 + 40 * q^43 - 11 * q^44 + 2 * q^45 + 6 * q^46 - 12 * q^47 + q^48 - q^49 - q^50 - q^51 - 2 * q^52 + 14 * q^53 + 4 * q^54 - 8 * q^55 - 4 * q^56 + 6 * q^57 - 8 * q^58 + 8 * q^59 + 3 * q^60 + 24 * q^61 + 6 * q^62 - q^63 - q^64 + 4 * q^65 + 4 * q^66 - 4 * q^67 - 4 * q^68 - 9 * q^69 - 2 * q^70 + 8 * q^71 + q^72 + 24 * q^73 + 8 * q^74 - q^75 - 6 * q^76 - 11 * q^77 + 8 * q^78 + 2 * q^80 - q^81 - q^82 - 10 * q^83 + q^84 + 3 * q^85 + 10 * q^86 - 8 * q^87 - 9 * q^88 - 14 * q^89 + 3 * q^90 - 2 * q^91 + 9 * q^92 + q^93 - 8 * q^94 - 3 * q^95 - q^96 - 26 * 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 −1.30902 + 0.951057i −0.809017 + 0.587785i 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i −1.61803
295.1 −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.190983 0.587785i 0.309017 + 0.951057i −0.809017 0.587785i 0.809017 0.587785i 0.309017 0.951057i 0.618034
379.1 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.190983 + 0.587785i 0.309017 0.951057i −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i 0.618034
421.1 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −1.30902 0.951057i −0.809017 0.587785i 0.309017 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i −1.61803
 $$n$$: 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.d 4
11.c even 5 1 inner 462.2.j.d 4
11.c even 5 1 5082.2.a.bf 2
11.d odd 10 1 5082.2.a.bp 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 3T_{5}^{3} + 4T_{5}^{2} + 2T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$7$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$11$ $$T^{4} + T^{3} + 21 T^{2} + 11 T + 121$$
$13$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$17$ $$T^{4} + 4 T^{3} + 6 T^{2} - T + 1$$
$19$ $$T^{4} - 9 T^{3} + 36 T^{2} - 54 T + 81$$
$23$ $$(T^{2} + 3 T - 9)^{2}$$
$29$ $$T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256$$
$31$ $$T^{4} + T^{3} + 16 T^{2} + 66 T + 121$$
$37$ $$T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121$$
$41$ $$T^{4} + 4 T^{3} + 6 T^{2} - T + 1$$
$43$ $$(T - 10)^{4}$$
$47$ $$T^{4} + 12 T^{3} + 64 T^{2} + \cdots + 256$$
$53$ $$T^{4} - 14 T^{3} + 76 T^{2} - 24 T + 16$$
$59$ $$T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16$$
$61$ $$T^{4} - 24 T^{3} + 376 T^{2} + \cdots + 15376$$
$67$ $$(T^{2} + 2 T - 124)^{2}$$
$71$ $$T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256$$
$73$ $$T^{4} - 24 T^{3} + 376 T^{2} + \cdots + 15376$$
$79$ $$T^{4} + 160 T^{2} + 1600 T + 6400$$
$83$ $$T^{4} + 10 T^{3} + 40 T^{2} + \cdots + 400$$
$89$ $$(T^{2} + 7 T - 139)^{2}$$
$97$ $$T^{4} + 26 T^{3} + 376 T^{2} + \cdots + 15376$$