# Properties

 Label 420.2.n.b Level $420$ Weight $2$ Character orbit 420.n Analytic conductor $3.354$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 2 q^{4} + 6 q^{6} + 4 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q + 2 q^{4} + 6 q^{6} + 4 q^{9} + 2 q^{10} + 16 q^{12} + 2 q^{14} + 6 q^{16} - 24 q^{18} - 8 q^{20} + 8 q^{22} + 14 q^{24} - 24 q^{25} + 20 q^{26} - 8 q^{28} - 8 q^{30} - 20 q^{32} + 16 q^{33} - 16 q^{34} - 24 q^{35} + 30 q^{36} + 60 q^{38} + 12 q^{39} - 14 q^{40} - 8 q^{42} - 24 q^{44} - 12 q^{46} - 8 q^{47} + 36 q^{48} - 24 q^{49} - 36 q^{51} + 20 q^{52} - 38 q^{54} - 14 q^{56} - 24 q^{57} + 44 q^{58} + 8 q^{59} + 14 q^{60} + 16 q^{61} + 28 q^{62} - 22 q^{64} - 12 q^{66} - 32 q^{68} - 72 q^{71} + 56 q^{72} - 24 q^{73} + 64 q^{74} + 48 q^{76} - 92 q^{78} - 20 q^{81} - 16 q^{82} + 40 q^{83} + 14 q^{84} - 16 q^{85} + 40 q^{86} + 80 q^{87} - 12 q^{88} - 10 q^{90} - 108 q^{92} - 48 q^{93} - 36 q^{94} + 34 q^{96} + 24 q^{97} - 84 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1 −1.41258 0.0678578i 1.66901 0.463024i 1.99079 + 0.191710i 1.00000i −2.38904 + 0.540805i 1.00000i −2.79915 0.405897i 2.57122 1.54559i 0.0678578 1.41258i
71.2 −1.41258 + 0.0678578i 1.66901 + 0.463024i 1.99079 0.191710i 1.00000i −2.38904 0.540805i 1.00000i −2.79915 + 0.405897i 2.57122 + 1.54559i 0.0678578 + 1.41258i
71.3 −1.38203 0.299986i −0.491627 + 1.66081i 1.82002 + 0.829179i 1.00000i 1.17766 2.14781i 1.00000i −2.26658 1.69193i −2.51661 1.63300i −0.299986 + 1.38203i
71.4 −1.38203 + 0.299986i −0.491627 1.66081i 1.82002 0.829179i 1.00000i 1.17766 + 2.14781i 1.00000i −2.26658 + 1.69193i −2.51661 + 1.63300i −0.299986 1.38203i
71.5 −1.09313 0.897253i −1.62224 + 0.606916i 0.389874 + 1.96163i 1.00000i 2.31788 + 0.792118i 1.00000i 1.33390 2.49414i 2.26331 1.96912i 0.897253 1.09313i
71.6 −1.09313 + 0.897253i −1.62224 0.606916i 0.389874 1.96163i 1.00000i 2.31788 0.792118i 1.00000i 1.33390 + 2.49414i 2.26331 + 1.96912i 0.897253 + 1.09313i
71.7 −0.859821 1.12281i −1.72451 0.161413i −0.521414 + 1.93084i 1.00000i 1.30154 + 2.07509i 1.00000i 2.61629 1.07472i 2.94789 + 0.556719i −1.12281 + 0.859821i
71.8 −0.859821 + 1.12281i −1.72451 + 0.161413i −0.521414 1.93084i 1.00000i 1.30154 2.07509i 1.00000i 2.61629 + 1.07472i 2.94789 0.556719i −1.12281 0.859821i
71.9 −0.637711 1.26227i 1.30156 1.14278i −1.18665 + 1.60993i 1.00000i −2.27252 0.914147i 1.00000i 2.78890 + 0.471201i 0.388091 2.97479i 1.26227 0.637711i
71.10 −0.637711 + 1.26227i 1.30156 + 1.14278i −1.18665 1.60993i 1.00000i −2.27252 + 0.914147i 1.00000i 2.78890 0.471201i 0.388091 + 2.97479i 1.26227 + 0.637711i
71.11 −0.0797590 1.41196i −1.52443 0.822258i −1.98728 + 0.225233i 1.00000i −1.03941 + 2.21802i 1.00000i 0.476524 + 2.78800i 1.64778 + 2.50695i 1.41196 0.0797590i
71.12 −0.0797590 + 1.41196i −1.52443 + 0.822258i −1.98728 0.225233i 1.00000i −1.03941 2.21802i 1.00000i 0.476524 2.78800i 1.64778 2.50695i 1.41196 + 0.0797590i
71.13 0.269719 1.38825i −0.0343495 + 1.73171i −1.85450 0.748876i 1.00000i 2.39479 + 0.514760i 1.00000i −1.53983 + 2.37254i −2.99764 0.118967i 1.38825 + 0.269719i
71.14 0.269719 + 1.38825i −0.0343495 1.73171i −1.85450 + 0.748876i 1.00000i 2.39479 0.514760i 1.00000i −1.53983 2.37254i −2.99764 + 0.118967i 1.38825 0.269719i
71.15 0.275453 1.38713i 1.30386 1.14015i −1.84825 0.764177i 1.00000i −1.22239 2.12268i 1.00000i −1.56912 + 2.35327i 0.400103 2.97320i −1.38713 0.275453i
71.16 0.275453 + 1.38713i 1.30386 + 1.14015i −1.84825 + 0.764177i 1.00000i −1.22239 + 2.12268i 1.00000i −1.56912 2.35327i 0.400103 + 2.97320i −1.38713 + 0.275453i
71.17 1.09123 0.899566i −0.826488 1.52214i 0.381563 1.96327i 1.00000i −2.27116 0.917526i 1.00000i −1.34971 2.48561i −1.63383 + 2.51607i −0.899566 1.09123i
71.18 1.09123 + 0.899566i −0.826488 + 1.52214i 0.381563 + 1.96327i 1.00000i −2.27116 + 0.917526i 1.00000i −1.34971 + 2.48561i −1.63383 2.51607i −0.899566 + 1.09123i
71.19 1.19442 0.757207i −0.667879 + 1.59810i 0.853276 1.80884i 1.00000i 0.412368 + 2.41453i 1.00000i −0.350500 2.80663i −2.10788 2.13468i −0.757207 1.19442i
71.20 1.19442 + 0.757207i −0.667879 1.59810i 0.853276 + 1.80884i 1.00000i 0.412368 2.41453i 1.00000i −0.350500 + 2.80663i −2.10788 + 2.13468i −0.757207 + 1.19442i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.n.b yes 24
3.b odd 2 1 420.2.n.a 24
4.b odd 2 1 420.2.n.a 24
12.b even 2 1 inner 420.2.n.b yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.n.a 24 3.b odd 2 1
420.2.n.a 24 4.b odd 2 1
420.2.n.b yes 24 1.a even 1 1 trivial
420.2.n.b yes 24 12.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$.