# Properties

 Label 462.2.y.c Level $462$ Weight $2$ Character orbit 462.y Analytic conductor $3.689$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.y (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$5$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 5q^{2} + 5q^{3} + 5q^{4} + q^{5} + 10q^{6} + 9q^{7} + 10q^{8} + 5q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 5q^{2} + 5q^{3} + 5q^{4} + q^{5} + 10q^{6} + 9q^{7} + 10q^{8} + 5q^{9} + 4q^{10} + 2q^{11} - 20q^{12} - 2q^{13} - 3q^{14} - 2q^{15} + 5q^{16} + 6q^{17} - 5q^{18} - q^{19} - 2q^{20} - 2q^{21} + 4q^{22} - 20q^{23} - 5q^{24} + 16q^{25} + 4q^{26} - 10q^{27} + 13q^{28} + 16q^{29} - q^{30} + 30q^{31} + 20q^{32} + 2q^{33} + 32q^{34} + 3q^{35} - 10q^{36} + 19q^{37} + 6q^{38} + q^{39} - q^{40} - 12q^{41} + 11q^{42} - 36q^{43} - 3q^{44} - 4q^{45} + 10q^{46} + 25q^{47} - 10q^{48} + 29q^{49} + 12q^{50} - 14q^{51} - 4q^{52} - 18q^{53} + 20q^{54} + 12q^{55} - 4q^{56} + 12q^{57} + 8q^{58} - 14q^{59} + q^{60} + 35q^{61} - 40q^{62} + 3q^{63} - 10q^{64} - 2q^{65} + 3q^{66} - 24q^{67} + 6q^{68} - 40q^{69} - 32q^{70} - 34q^{71} - 5q^{72} - 39q^{73} - 19q^{74} + 6q^{75} - 28q^{76} - 31q^{77} + 12q^{78} - 17q^{79} + q^{80} + 5q^{81} + 4q^{82} - 24q^{83} + 3q^{84} - 112q^{85} + 7q^{86} + 2q^{87} - 2q^{88} - 38q^{89} + 2q^{90} - 36q^{91} - 40q^{92} + 30q^{93} - 10q^{94} - 42q^{95} - 5q^{96} + 4q^{97} - 50q^{98} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 −0.913545 0.406737i −0.978148 0.207912i 0.669131 + 0.743145i −0.435132 + 4.14000i 0.809017 + 0.587785i 2.64557 0.0306495i −0.309017 0.951057i 0.913545 + 0.406737i 2.08140 3.60509i
25.2 −0.913545 0.406737i −0.978148 0.207912i 0.669131 + 0.743145i −0.168631 + 1.60442i 0.809017 + 0.587785i −1.86877 1.87289i −0.309017 0.951057i 0.913545 + 0.406737i 0.806628 1.39712i
25.3 −0.913545 0.406737i −0.978148 0.207912i 0.669131 + 0.743145i −0.0207786 + 0.197695i 0.809017 + 0.587785i −2.07849 + 1.63704i −0.309017 0.951057i 0.913545 + 0.406737i 0.0993921 0.172152i
25.4 −0.913545 0.406737i −0.978148 0.207912i 0.669131 + 0.743145i 0.171980 1.63628i 0.809017 + 0.587785i 2.59574 + 0.512007i −0.309017 0.951057i 0.913545 + 0.406737i −0.822649 + 1.42487i
25.5 −0.913545 0.406737i −0.978148 0.207912i 0.669131 + 0.743145i 0.348032 3.31131i 0.809017 + 0.587785i 1.39315 2.24925i −0.309017 0.951057i 0.913545 + 0.406737i −1.66477 + 2.88347i
37.1 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 0.743145i −0.435132 4.14000i 0.809017 0.587785i 2.64557 + 0.0306495i −0.309017 + 0.951057i 0.913545 0.406737i 2.08140 + 3.60509i
37.2 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 0.743145i −0.168631 1.60442i 0.809017 0.587785i −1.86877 + 1.87289i −0.309017 + 0.951057i 0.913545 0.406737i 0.806628 + 1.39712i
37.3 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 0.743145i −0.0207786 0.197695i 0.809017 0.587785i −2.07849 1.63704i −0.309017 + 0.951057i 0.913545 0.406737i 0.0993921 + 0.172152i
37.4 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 0.743145i 0.171980 + 1.63628i 0.809017 0.587785i 2.59574 0.512007i −0.309017 + 0.951057i 0.913545 0.406737i −0.822649 1.42487i
37.5 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 0.743145i 0.348032 + 3.31131i 0.809017 0.587785i 1.39315 + 2.24925i −0.309017 + 0.951057i 0.913545 0.406737i −1.66477 2.88347i
163.1 −0.669131 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i −3.51246 0.746597i −0.309017 0.951057i 2.57655 + 0.601166i 0.809017 0.587785i 0.669131 + 0.743145i 1.79547 + 3.10984i
163.2 −0.669131 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i −2.08985 0.444211i −0.309017 0.951057i −2.27148 + 1.35660i 0.809017 0.587785i 0.669131 + 0.743145i 1.06827 + 1.85029i
163.3 −0.669131 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i −1.13392 0.241022i −0.309017 0.951057i −1.18884 2.36361i 0.809017 0.587785i 0.669131 + 0.743145i 0.579626 + 1.00394i
163.4 −0.669131 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 1.87534 + 0.398616i −0.309017 0.951057i 1.79574 + 1.94302i 0.809017 0.587785i 0.669131 + 0.743145i −0.958617 1.66037i
163.5 −0.669131 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 3.88274 + 0.825303i −0.309017 0.951057i 1.63079 2.08339i 0.809017 0.587785i 0.669131 + 0.743145i −1.98474 3.43768i
235.1 0.104528 0.994522i 0.669131 + 0.743145i −0.978148 0.207912i −3.04169 1.35425i 0.809017 0.587785i 0.194999 + 2.63856i −0.309017 + 0.951057i −0.104528 + 0.994522i −1.66477 + 2.88347i
235.2 0.104528 0.994522i 0.669131 + 0.743145i −0.978148 0.207912i −1.50305 0.669203i 0.809017 0.587785i −2.40095 + 1.11151i −0.309017 + 0.951057i −0.104528 + 0.994522i −0.822649 + 1.42487i
235.3 0.104528 0.994522i 0.669131 + 0.743145i −0.978148 0.207912i 0.181598 + 0.0808528i 0.809017 0.587785i 0.719303 2.54610i −0.309017 + 0.951057i −0.104528 + 0.994522i 0.0993921 0.172152i
235.4 0.104528 0.994522i 0.669131 + 0.743145i −0.978148 0.207912i 1.47378 + 0.656170i 0.809017 0.587785i 2.61272 + 0.416765i −0.309017 + 0.951057i −0.104528 + 0.994522i 0.806628 1.39712i
235.5 0.104528 0.994522i 0.669131 + 0.743145i −0.978148 0.207912i 3.80291 + 1.69317i 0.809017 0.587785i −2.12230 + 1.57983i −0.309017 + 0.951057i −0.104528 + 0.994522i 2.08140 3.60509i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 445.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.y.c 40
7.c even 3 1 inner 462.2.y.c 40
11.c even 5 1 inner 462.2.y.c 40
77.m even 15 1 inner 462.2.y.c 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.y.c 40 1.a even 1 1 trivial
462.2.y.c 40 7.c even 3 1 inner
462.2.y.c 40 11.c even 5 1 inner
462.2.y.c 40 77.m even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$17\!\cdots\!31$$$$T_{5}^{14} + 168944203314 T_{5}^{13} -$$$$31\!\cdots\!75$$$$T_{5}^{12} + 631098037422 T_{5}^{11} -$$$$16\!\cdots\!85$$$$T_{5}^{10} -$$$$23\!\cdots\!79$$$$T_{5}^{9} +$$$$23\!\cdots\!91$$$$T_{5}^{8} + 123470855145 T_{5}^{7} -$$$$26\!\cdots\!15$$$$T_{5}^{6} +$$$$31\!\cdots\!00$$$$T_{5}^{5} +$$$$29\!\cdots\!50$$$$T_{5}^{4} -$$$$10\!\cdots\!75$$$$T_{5}^{3} +$$$$21\!\cdots\!50$$$$T_{5}^{2} - 453008345625 T_{5} + 60037250625$$">$$T_{5}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.