# Properties

 Label 462.2.j.b 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: $$x^{4} - x^{3} + x^{2} - x + 1$$ 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.

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

## 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 0.309017 0.224514i 0.809017 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i −0.381966
295.1 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.809017 2.48990i −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i −2.61803
379.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.809017 + 2.48990i −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i −2.61803
421.1 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.309017 + 0.224514i 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.381966
 $$n$$: 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.b 4
11.c even 5 1 inner 462.2.j.b 4
11.c even 5 1 5082.2.a.bn 2
11.d odd 10 1 5082.2.a.bd 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + T_{5}^{3} + 6 T_{5}^{2} - 4 T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$1 - 4 T + 6 T^{2} + T^{3} + T^{4}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$121 - 11 T - 9 T^{2} - T^{3} + T^{4}$$
$13$ $$16 + 24 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$1 + T + 6 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$81 - 108 T + 54 T^{2} + 3 T^{3} + T^{4}$$
$23$ $$( 19 - 9 T + T^{2} )^{2}$$
$29$ $$16 + 24 T + 76 T^{2} + 14 T^{3} + T^{4}$$
$31$ $$81 - 54 T + 36 T^{2} - 9 T^{3} + T^{4}$$
$37$ $$81 - 81 T + 36 T^{2} - 6 T^{3} + T^{4}$$
$41$ $$841 - 87 T + 34 T^{2} - 8 T^{3} + T^{4}$$
$43$ $$( -4 + 2 T + T^{2} )^{2}$$
$47$ $$16 - 24 T + 76 T^{2} - 14 T^{3} + T^{4}$$
$53$ $$1296 - 432 T + 144 T^{2} - 18 T^{3} + T^{4}$$
$59$ $$400 + 40 T^{2} - 10 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -4 + T )^{4}$$
$71$ $$20736 - 1728 T + 144 T^{2} - 12 T^{3} + T^{4}$$
$73$ $$400 + 400 T + 160 T^{2} + 10 T^{3} + T^{4}$$
$79$ $$16 + 24 T + 76 T^{2} + 14 T^{3} + T^{4}$$
$83$ $$1936 - 1144 T + 276 T^{2} - 14 T^{3} + T^{4}$$
$89$ $$( -19 + 13 T + T^{2} )^{2}$$
$97$ $$16 + 56 T + 76 T^{2} + 6 T^{3} + T^{4}$$