# Properties

 Label 420.2.i.a Level $420$ Weight $2$ Character orbit 420.i Analytic conductor $3.354$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$48$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 48 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48 q - 48 q^{9} + 20 q^{14} - 16 q^{16} + 8 q^{25} - 16 q^{30} - 40 q^{44} + 16 q^{46} - 16 q^{49} + 48 q^{50} + 28 q^{56} - 32 q^{60} - 112 q^{74} + 48 q^{81} - 28 q^{84} + 56 q^{85} + 8 q^{86} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1 −1.36843 0.356944i 1.00000i 1.74518 + 0.976904i −1.65234 + 1.50657i −0.356944 + 1.36843i −2.07310 + 1.64386i −2.03945 1.95975i −1.00000 2.79887 1.47184i
139.2 −1.36843 0.356944i 1.00000i 1.74518 + 0.976904i 1.65234 1.50657i 0.356944 1.36843i −2.07310 1.64386i −2.03945 1.95975i −1.00000 −2.79887 + 1.47184i
139.3 −1.36843 + 0.356944i 1.00000i 1.74518 0.976904i 1.65234 + 1.50657i 0.356944 + 1.36843i −2.07310 + 1.64386i −2.03945 + 1.95975i −1.00000 −2.79887 1.47184i
139.4 −1.36843 + 0.356944i 1.00000i 1.74518 0.976904i −1.65234 1.50657i −0.356944 1.36843i −2.07310 1.64386i −2.03945 + 1.95975i −1.00000 2.79887 + 1.47184i
139.5 −1.33186 0.475542i 1.00000i 1.54772 + 1.26671i 2.22087 + 0.260295i −0.475542 + 1.33186i 1.17807 2.36900i −1.45898 2.42309i −1.00000 −2.83411 1.40279i
139.6 −1.33186 0.475542i 1.00000i 1.54772 + 1.26671i −2.22087 0.260295i 0.475542 1.33186i 1.17807 + 2.36900i −1.45898 2.42309i −1.00000 2.83411 + 1.40279i
139.7 −1.33186 + 0.475542i 1.00000i 1.54772 1.26671i −2.22087 + 0.260295i 0.475542 + 1.33186i 1.17807 2.36900i −1.45898 + 2.42309i −1.00000 2.83411 1.40279i
139.8 −1.33186 + 0.475542i 1.00000i 1.54772 1.26671i 2.22087 0.260295i −0.475542 1.33186i 1.17807 + 2.36900i −1.45898 + 2.42309i −1.00000 −2.83411 + 1.40279i
139.9 −1.08769 0.903842i 1.00000i 0.366139 + 1.96620i −0.660150 2.13640i −0.903842 + 1.08769i −2.64346 0.110159i 1.37889 2.46955i −1.00000 −1.21293 + 2.92041i
139.10 −1.08769 0.903842i 1.00000i 0.366139 + 1.96620i 0.660150 + 2.13640i 0.903842 1.08769i −2.64346 + 0.110159i 1.37889 2.46955i −1.00000 1.21293 2.92041i
139.11 −1.08769 + 0.903842i 1.00000i 0.366139 1.96620i 0.660150 2.13640i 0.903842 + 1.08769i −2.64346 0.110159i 1.37889 + 2.46955i −1.00000 1.21293 + 2.92041i
139.12 −1.08769 + 0.903842i 1.00000i 0.366139 1.96620i −0.660150 + 2.13640i −0.903842 1.08769i −2.64346 + 0.110159i 1.37889 + 2.46955i −1.00000 −1.21293 2.92041i
139.13 −0.846354 1.13300i 1.00000i −0.567368 + 1.91784i −2.22361 + 0.235752i −1.13300 + 0.846354i 2.54904 + 0.708793i 2.65310 0.980342i −1.00000 2.14906 + 2.31981i
139.14 −0.846354 1.13300i 1.00000i −0.567368 + 1.91784i 2.22361 0.235752i 1.13300 0.846354i 2.54904 0.708793i 2.65310 0.980342i −1.00000 −2.14906 2.31981i
139.15 −0.846354 + 1.13300i 1.00000i −0.567368 1.91784i 2.22361 + 0.235752i 1.13300 + 0.846354i 2.54904 + 0.708793i 2.65310 + 0.980342i −1.00000 −2.14906 + 2.31981i
139.16 −0.846354 + 1.13300i 1.00000i −0.567368 1.91784i −2.22361 0.235752i −1.13300 0.846354i 2.54904 0.708793i 2.65310 + 0.980342i −1.00000 2.14906 2.31981i
139.17 −0.654401 1.25370i 1.00000i −1.14352 + 1.64084i −0.206931 + 2.22647i −1.25370 + 0.654401i −0.859661 2.50220i 2.80544 + 0.359858i −1.00000 2.92674 1.19758i
139.18 −0.654401 1.25370i 1.00000i −1.14352 + 1.64084i 0.206931 2.22647i 1.25370 0.654401i −0.859661 + 2.50220i 2.80544 + 0.359858i −1.00000 −2.92674 + 1.19758i
139.19 −0.654401 + 1.25370i 1.00000i −1.14352 1.64084i 0.206931 + 2.22647i 1.25370 + 0.654401i −0.859661 2.50220i 2.80544 0.359858i −1.00000 −2.92674 1.19758i
139.20 −0.654401 + 1.25370i 1.00000i −1.14352 1.64084i −0.206931 2.22647i −1.25370 0.654401i −0.859661 + 2.50220i 2.80544 0.359858i −1.00000 2.92674 + 1.19758i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c 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.i.a 48
4.b odd 2 1 inner 420.2.i.a 48
5.b even 2 1 inner 420.2.i.a 48
7.b odd 2 1 inner 420.2.i.a 48
20.d odd 2 1 inner 420.2.i.a 48
28.d even 2 1 inner 420.2.i.a 48
35.c odd 2 1 inner 420.2.i.a 48
140.c even 2 1 inner 420.2.i.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.i.a 48 1.a even 1 1 trivial
420.2.i.a 48 4.b odd 2 1 inner
420.2.i.a 48 5.b even 2 1 inner
420.2.i.a 48 7.b odd 2 1 inner
420.2.i.a 48 20.d odd 2 1 inner
420.2.i.a 48 28.d even 2 1 inner
420.2.i.a 48 35.c odd 2 1 inner
420.2.i.a 48 140.c even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(420, [\chi])$$.