# Properties

 Label 280.2.bo.a Level $280$ Weight $2$ Character orbit 280.bo Analytic conductor $2.236$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.bo (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 4 q^{7}+O(q^{10})$$ 48 * q - 4 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 4 q^{7} + 4 q^{11} + 8 q^{15} - 4 q^{21} - 4 q^{23} - 8 q^{25} - 36 q^{33} + 24 q^{35} + 8 q^{37} - 16 q^{43} + 48 q^{45} + 24 q^{51} + 16 q^{53} - 96 q^{57} - 36 q^{61} - 68 q^{63} + 12 q^{65} - 16 q^{67} - 64 q^{71} - 48 q^{73} - 48 q^{75} + 4 q^{77} - 40 q^{85} - 12 q^{87} - 80 q^{91} + 24 q^{95}+O(q^{100})$$ 48 * q - 4 * q^7 + 4 * q^11 + 8 * q^15 - 4 * q^21 - 4 * q^23 - 8 * q^25 - 36 * q^33 + 24 * q^35 + 8 * q^37 - 16 * q^43 + 48 * q^45 + 24 * q^51 + 16 * q^53 - 96 * q^57 - 36 * q^61 - 68 * q^63 + 12 * q^65 - 16 * q^67 - 64 * q^71 - 48 * q^73 - 48 * q^75 + 4 * q^77 - 40 * q^85 - 12 * q^87 - 80 * q^91 + 24 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −2.75887 + 0.739238i 0 −1.04631 + 1.97617i 0 −1.91229 1.82843i 0 4.46683 2.57893i 0
17.2 0 −2.30560 + 0.617784i 0 1.37859 1.76054i 0 −0.755351 + 2.53563i 0 2.33607 1.34873i 0
17.3 0 −2.00047 + 0.536025i 0 1.38383 + 1.75642i 0 1.24986 2.33192i 0 1.11649 0.644605i 0
17.4 0 −1.35944 + 0.364260i 0 −2.13431 0.666887i 0 2.59782 0.501338i 0 −0.882692 + 0.509622i 0
17.5 0 −1.16710 + 0.312724i 0 −0.121837 2.23275i 0 −2.59977 + 0.491095i 0 −1.33374 + 0.770036i 0
17.6 0 −0.0749389 + 0.0200798i 0 −0.965007 + 2.01712i 0 1.45892 + 2.20716i 0 −2.59286 + 1.49699i 0
17.7 0 0.120281 0.0322291i 0 −2.04377 + 0.907205i 0 −2.46167 + 0.969635i 0 −2.58465 + 1.49225i 0
17.8 0 0.280752 0.0752271i 0 2.21121 0.332503i 0 1.13104 2.39181i 0 −2.52491 + 1.45776i 0
17.9 0 1.66856 0.447090i 0 2.21600 + 0.298941i 0 0.900705 + 2.48772i 0 −0.0138674 + 0.00800633i 0
17.10 0 2.19810 0.588978i 0 −1.19880 1.88756i 0 1.40409 + 2.24244i 0 1.88665 1.08926i 0
17.11 0 2.20364 0.590462i 0 −0.352053 2.20818i 0 −1.22435 2.34541i 0 1.90929 1.10233i 0
17.12 0 3.19510 0.856125i 0 0.672461 + 2.13256i 0 −2.52104 0.802724i 0 6.87765 3.97081i 0
33.1 0 −2.75887 0.739238i 0 −1.04631 1.97617i 0 −1.91229 + 1.82843i 0 4.46683 + 2.57893i 0
33.2 0 −2.30560 0.617784i 0 1.37859 + 1.76054i 0 −0.755351 2.53563i 0 2.33607 + 1.34873i 0
33.3 0 −2.00047 0.536025i 0 1.38383 1.75642i 0 1.24986 + 2.33192i 0 1.11649 + 0.644605i 0
33.4 0 −1.35944 0.364260i 0 −2.13431 + 0.666887i 0 2.59782 + 0.501338i 0 −0.882692 0.509622i 0
33.5 0 −1.16710 0.312724i 0 −0.121837 + 2.23275i 0 −2.59977 0.491095i 0 −1.33374 0.770036i 0
33.6 0 −0.0749389 0.0200798i 0 −0.965007 2.01712i 0 1.45892 2.20716i 0 −2.59286 1.49699i 0
33.7 0 0.120281 + 0.0322291i 0 −2.04377 0.907205i 0 −2.46167 0.969635i 0 −2.58465 1.49225i 0
33.8 0 0.280752 + 0.0752271i 0 2.21121 + 0.332503i 0 1.13104 + 2.39181i 0 −2.52491 1.45776i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bo.a 48
4.b odd 2 1 560.2.ci.e 48
5.c odd 4 1 inner 280.2.bo.a 48
7.d odd 6 1 inner 280.2.bo.a 48
20.e even 4 1 560.2.ci.e 48
28.f even 6 1 560.2.ci.e 48
35.k even 12 1 inner 280.2.bo.a 48
140.x odd 12 1 560.2.ci.e 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bo.a 48 1.a even 1 1 trivial
280.2.bo.a 48 5.c odd 4 1 inner
280.2.bo.a 48 7.d odd 6 1 inner
280.2.bo.a 48 35.k even 12 1 inner
560.2.ci.e 48 4.b odd 2 1
560.2.ci.e 48 20.e even 4 1
560.2.ci.e 48 28.f even 6 1
560.2.ci.e 48 140.x odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(280, [\chi])$$.