# Properties

 Label 462.2.w.b 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.

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

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 0.809017 + 0.587785i −1.67366 0.445927i 0.309017 + 0.951057i 1.77599 + 2.44445i −1.09191 1.34452i −0.951057 + 0.309017i −0.309017 + 0.951057i 2.60230 + 1.49266i 3.02150i
29.2 0.809017 + 0.587785i −1.57790 + 0.714295i 0.309017 + 0.951057i −1.72358 2.37230i −1.69640 0.349593i 0.951057 0.309017i −0.309017 + 0.951057i 1.97957 2.25418i 2.93233i
29.3 0.809017 + 0.587785i −1.35082 + 1.08410i 0.309017 + 0.951057i 1.96876 + 2.70976i −1.73006 + 0.0830614i 0.951057 0.309017i −0.309017 + 0.951057i 0.649451 2.92886i 3.34945i
29.4 0.809017 + 0.587785i −1.32326 1.11758i 0.309017 + 0.951057i −2.46323 3.39035i −0.413643 1.68193i −0.951057 + 0.309017i −0.309017 + 0.951057i 0.502035 + 2.95770i 4.19070i
29.5 0.809017 + 0.587785i −0.979414 1.42855i 0.309017 + 0.951057i 0.571630 + 0.786781i 0.0473160 1.73140i 0.951057 0.309017i −0.309017 + 0.951057i −1.08149 + 2.79828i 0.972514i
29.6 0.809017 + 0.587785i 0.407282 + 1.68348i 0.309017 + 0.951057i 1.88925 + 2.60033i −0.660029 + 1.60136i −0.951057 + 0.309017i −0.309017 + 0.951057i −2.66824 + 1.37131i 3.21418i
29.7 0.809017 + 0.587785i 0.677330 + 1.59412i 0.309017 + 0.951057i −0.456676 0.628560i −0.389030 + 1.68780i 0.951057 0.309017i −0.309017 + 0.951057i −2.08245 + 2.15949i 0.776943i
29.8 0.809017 + 0.587785i 0.794507 1.53908i 0.309017 + 0.951057i −1.43689 1.97771i 1.54742 0.778140i −0.951057 + 0.309017i −0.309017 + 0.951057i −1.73752 2.44562i 2.44458i
29.9 0.809017 + 0.587785i 1.39186 1.03088i 0.309017 + 0.951057i 1.54008 + 2.11973i 1.73198 0.0158825i −0.951057 + 0.309017i −0.309017 + 0.951057i 0.874572 2.86969i 2.62013i
29.10 0.809017 + 0.587785i 1.60305 0.655932i 0.309017 + 0.951057i −2.04373 2.81296i 1.68244 + 0.411586i 0.951057 0.309017i −0.309017 + 0.951057i 2.13951 2.10298i 3.47700i
29.11 0.809017 + 0.587785i 1.60909 + 0.640962i 0.309017 + 0.951057i −0.354145 0.487439i 0.925033 + 1.46435i −0.951057 + 0.309017i −0.309017 + 0.951057i 2.17834 + 2.06273i 0.602508i
29.12 0.809017 + 0.587785i 1.65802 + 0.500979i 0.309017 + 0.951057i 0.732542 + 1.00826i 1.04690 + 1.37986i 0.951057 0.309017i −0.309017 + 0.951057i 2.49804 + 1.66126i 1.24628i
239.1 0.809017 0.587785i −1.67366 + 0.445927i 0.309017 0.951057i 1.77599 2.44445i −1.09191 + 1.34452i −0.951057 0.309017i −0.309017 0.951057i 2.60230 1.49266i 3.02150i
239.2 0.809017 0.587785i −1.57790 0.714295i 0.309017 0.951057i −1.72358 + 2.37230i −1.69640 + 0.349593i 0.951057 + 0.309017i −0.309017 0.951057i 1.97957 + 2.25418i 2.93233i
239.3 0.809017 0.587785i −1.35082 1.08410i 0.309017 0.951057i 1.96876 2.70976i −1.73006 0.0830614i 0.951057 + 0.309017i −0.309017 0.951057i 0.649451 + 2.92886i 3.34945i
239.4 0.809017 0.587785i −1.32326 + 1.11758i 0.309017 0.951057i −2.46323 + 3.39035i −0.413643 + 1.68193i −0.951057 0.309017i −0.309017 0.951057i 0.502035 2.95770i 4.19070i
239.5 0.809017 0.587785i −0.979414 + 1.42855i 0.309017 0.951057i 0.571630 0.786781i 0.0473160 + 1.73140i 0.951057 + 0.309017i −0.309017 0.951057i −1.08149 2.79828i 0.972514i
239.6 0.809017 0.587785i 0.407282 1.68348i 0.309017 0.951057i 1.88925 2.60033i −0.660029 1.60136i −0.951057 0.309017i −0.309017 0.951057i −2.66824 1.37131i 3.21418i
239.7 0.809017 0.587785i 0.677330 1.59412i 0.309017 0.951057i −0.456676 + 0.628560i −0.389030 1.68780i 0.951057 + 0.309017i −0.309017 0.951057i −2.08245 2.15949i 0.776943i
239.8 0.809017 0.587785i 0.794507 + 1.53908i 0.309017 0.951057i −1.43689 + 1.97771i 1.54742 + 0.778140i −0.951057 0.309017i −0.309017 0.951057i −1.73752 + 2.44562i 2.44458i
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.b yes 48
3.b odd 2 1 462.2.w.a 48
11.d odd 10 1 462.2.w.a 48
33.f even 10 1 inner 462.2.w.b yes 48

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$34\!\cdots\!30$$$$T_{5}^{26} +$$$$37\!\cdots\!00$$$$T_{5}^{25} +$$$$18\!\cdots\!66$$$$T_{5}^{24} -$$$$24\!\cdots\!40$$$$T_{5}^{23} -$$$$81\!\cdots\!48$$$$T_{5}^{22} +$$$$11\!\cdots\!40$$$$T_{5}^{21} +$$$$37\!\cdots\!22$$$$T_{5}^{20} -$$$$62\!\cdots\!40$$$$T_{5}^{19} -$$$$83\!\cdots\!20$$$$T_{5}^{18} +$$$$20\!\cdots\!80$$$$T_{5}^{17} +$$$$58\!\cdots\!61$$$$T_{5}^{16} -$$$$38\!\cdots\!40$$$$T_{5}^{15} +$$$$12\!\cdots\!21$$$$T_{5}^{14} +$$$$41\!\cdots\!80$$$$T_{5}^{13} -$$$$24\!\cdots\!10$$$$T_{5}^{12} -$$$$25\!\cdots\!00$$$$T_{5}^{11} +$$$$18\!\cdots\!88$$$$T_{5}^{10} +$$$$15\!\cdots\!00$$$$T_{5}^{9} -$$$$83\!\cdots\!75$$$$T_{5}^{8} -$$$$16\!\cdots\!60$$$$T_{5}^{7} +$$$$10\!\cdots\!24$$$$T_{5}^{6} +$$$$30\!\cdots\!40$$$$T_{5}^{5} +$$$$37\!\cdots\!68$$$$T_{5}^{4} -$$$$26\!\cdots\!60$$$$T_{5}^{3} +$$$$32\!\cdots\!92$$$$T_{5}^{2} +$$$$27\!\cdots\!40$$$$T_{5} +$$$$29\!\cdots\!36$$">$$T_{5}^{48} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.