# Properties

 Label 462.2.w.a Level $462$ Weight $2$ Character orbit 462.w Analytic conductor $3.689$ Analytic rank $0$ Dimension $48$ 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.w (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

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

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −0.809017 0.587785i −1.67853 0.427250i 0.309017 + 0.951057i 0.354145 + 0.487439i 1.10683 + 1.33227i −0.951057 + 0.309017i 0.309017 0.951057i 2.63492 + 1.43430i 0.602508i
29.2 −0.809017 0.587785i −1.63583 0.569257i 0.309017 + 0.951057i −0.732542 1.00826i 0.988815 + 1.42206i 0.951057 0.309017i 0.309017 0.951057i 2.35189 + 1.86242i 1.24628i
29.3 −0.809017 0.587785i −1.48497 + 0.891547i 0.309017 + 0.951057i 0.456676 + 0.628560i 1.72541 + 0.151568i 0.951057 0.309017i 0.309017 0.951057i 1.41029 2.64785i 0.776943i
29.4 −0.809017 0.587785i −1.31903 + 1.12257i 0.309017 + 0.951057i −1.88925 2.60033i 1.72695 0.132877i −0.951057 + 0.309017i 0.309017 0.951057i 0.479659 2.96141i 3.21418i
29.5 −0.809017 0.587785i −0.911344 1.47291i 0.309017 + 0.951057i 2.04373 + 2.81296i −0.128460 + 1.72728i 0.951057 0.309017i 0.309017 0.951057i −1.33891 + 2.68465i 3.47700i
29.6 −0.809017 0.587785i −0.520106 1.65212i 0.309017 + 0.951057i −1.54008 2.11973i −0.550316 + 1.64230i −0.951057 + 0.309017i 0.309017 0.951057i −2.45898 + 1.71855i 2.62013i
29.7 −0.809017 0.587785i 0.261877 1.71214i 0.309017 + 0.951057i 1.43689 + 1.97771i −1.21823 + 1.23122i −0.951057 + 0.309017i 0.309017 0.951057i −2.86284 0.896741i 2.44458i
29.8 −0.809017 0.587785i 0.455621 + 1.67105i 0.309017 + 0.951057i −1.96876 2.70976i 0.613613 1.61972i 0.951057 0.309017i 0.309017 0.951057i −2.58482 + 1.52273i 3.34945i
29.9 −0.809017 0.587785i 0.856700 + 1.50535i 0.309017 + 0.951057i 1.72358 + 2.37230i 0.191735 1.72141i 0.951057 0.309017i 0.309017 0.951057i −1.53213 + 2.57926i 2.93233i
29.10 −0.809017 0.587785i 1.61613 + 0.622992i 0.309017 + 0.951057i −1.77599 2.44445i −0.941292 1.45395i −0.951057 + 0.309017i 0.309017 0.951057i 2.22376 + 2.01367i 3.02150i
29.11 −0.809017 0.587785i 1.63204 0.580034i 0.309017 + 0.951057i −0.571630 0.786781i −1.66128 0.490033i 0.951057 0.309017i 0.309017 0.951057i 2.32712 1.89328i 0.972514i
29.12 −0.809017 0.587785i 1.72744 0.126348i 0.309017 + 0.951057i 2.46323 + 3.39035i −1.47179 0.913144i −0.951057 + 0.309017i 0.309017 0.951057i 2.96807 0.436515i 4.19070i
239.1 −0.809017 + 0.587785i −1.67853 + 0.427250i 0.309017 0.951057i 0.354145 0.487439i 1.10683 1.33227i −0.951057 0.309017i 0.309017 + 0.951057i 2.63492 1.43430i 0.602508i
239.2 −0.809017 + 0.587785i −1.63583 + 0.569257i 0.309017 0.951057i −0.732542 + 1.00826i 0.988815 1.42206i 0.951057 + 0.309017i 0.309017 + 0.951057i 2.35189 1.86242i 1.24628i
239.3 −0.809017 + 0.587785i −1.48497 0.891547i 0.309017 0.951057i 0.456676 0.628560i 1.72541 0.151568i 0.951057 + 0.309017i 0.309017 + 0.951057i 1.41029 + 2.64785i 0.776943i
239.4 −0.809017 + 0.587785i −1.31903 1.12257i 0.309017 0.951057i −1.88925 + 2.60033i 1.72695 + 0.132877i −0.951057 0.309017i 0.309017 + 0.951057i 0.479659 + 2.96141i 3.21418i
239.5 −0.809017 + 0.587785i −0.911344 + 1.47291i 0.309017 0.951057i 2.04373 2.81296i −0.128460 1.72728i 0.951057 + 0.309017i 0.309017 + 0.951057i −1.33891 2.68465i 3.47700i
239.6 −0.809017 + 0.587785i −0.520106 + 1.65212i 0.309017 0.951057i −1.54008 + 2.11973i −0.550316 1.64230i −0.951057 0.309017i 0.309017 + 0.951057i −2.45898 1.71855i 2.62013i
239.7 −0.809017 + 0.587785i 0.261877 + 1.71214i 0.309017 0.951057i 1.43689 1.97771i −1.21823 1.23122i −0.951057 0.309017i 0.309017 + 0.951057i −2.86284 + 0.896741i 2.44458i
239.8 −0.809017 + 0.587785i 0.455621 1.67105i 0.309017 0.951057i −1.96876 + 2.70976i 0.613613 + 1.61972i 0.951057 + 0.309017i 0.309017 + 0.951057i −2.58482 1.52273i 3.34945i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 365.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.w.a 48
3.b odd 2 1 462.2.w.b yes 48
11.d odd 10 1 462.2.w.b yes 48
33.f even 10 1 inner 462.2.w.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.w.a 48 1.a even 1 1 trivial
462.2.w.a 48 33.f even 10 1 inner
462.2.w.b yes 48 3.b odd 2 1
462.2.w.b yes 48 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{48} - 39 T_{5}^{46} + 20 T_{5}^{45} + 976 T_{5}^{44} - 780 T_{5}^{43} - 19614 T_{5}^{42} + 14840 T_{5}^{41} + 373482 T_{5}^{40} - 144760 T_{5}^{39} - 5368055 T_{5}^{38} + 1415240 T_{5}^{37} + \cdots + 2900399739136$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.