# Properties

 Label 287.3.o.a Level 287 Weight 3 Character orbit 287.o Analytic conductor 7.820 Analytic rank 0 Dimension 216 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.o (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$216$$ Relative dimension: $$54$$ over $$\Q(\zeta_{10})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$216q - 6q^{2} - 110q^{4} - 3q^{7} - 2q^{8} - 648q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$216q - 6q^{2} - 110q^{4} - 3q^{7} - 2q^{8} - 648q^{9} - 22q^{11} + 82q^{14} + 36q^{15} - 182q^{16} + 24q^{18} + 131q^{21} - 38q^{22} - 66q^{23} + 244q^{25} + 56q^{28} - 230q^{29} - 184q^{30} + 56q^{32} - 120q^{35} + 100q^{37} + 148q^{39} - 164q^{42} - 44q^{43} + 448q^{44} - 260q^{46} - 151q^{49} - 116q^{50} + 114q^{51} + 388q^{53} + 77q^{56} + 180q^{57} + 222q^{58} - 382q^{60} - 29q^{63} - 82q^{64} + 636q^{65} + 426q^{67} - 527q^{70} + 360q^{71} - 466q^{72} - 236q^{74} + 34q^{77} - 150q^{78} - 96q^{79} + 1384q^{81} + 858q^{84} - 252q^{85} - 420q^{86} + 284q^{88} + 878q^{91} - 26q^{92} - 120q^{93} - 1406q^{95} + 641q^{98} - 398q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1 −1.19816 + 3.68755i 4.62584i −8.92636 6.48538i −1.55146 + 2.13541i 17.0580 + 5.54248i −0.575786 6.97628i 22.0631 16.0298i −12.3984 −6.01552 8.27966i
139.2 −1.19816 + 3.68755i 4.62584i −8.92636 6.48538i 1.55146 2.13541i −17.0580 5.54248i −3.63473 5.98237i 22.0631 16.0298i −12.3984 6.01552 + 8.27966i
139.3 −1.08729 + 3.34632i 0.232377i −6.77961 4.92568i 1.40348 1.93173i 0.777610 + 0.252661i −6.87163 + 1.33444i 12.4681 9.05858i 8.94600 4.93820 + 6.79685i
139.4 −1.08729 + 3.34632i 0.232377i −6.77961 4.92568i −1.40348 + 1.93173i −0.777610 0.252661i 6.34363 2.95945i 12.4681 9.05858i 8.94600 −4.93820 6.79685i
139.5 −1.08527 + 3.34011i 1.54191i −6.74248 4.89870i −3.83280 + 5.27539i 5.15016 + 1.67339i −0.829150 + 6.95072i 12.3145 8.94703i 6.62251 −13.4608 18.5272i
139.6 −1.08527 + 3.34011i 1.54191i −6.74248 4.89870i 3.83280 5.27539i −5.15016 1.67339i 4.75633 + 5.13589i 12.3145 8.94703i 6.62251 13.4608 + 18.5272i
139.7 −0.961384 + 2.95883i 3.57784i −4.59437 3.33801i 5.23870 7.21045i 10.5862 + 3.43967i −6.98907 + 0.391037i 4.22583 3.07025i −3.80091 16.2981 + 22.4324i
139.8 −0.961384 + 2.95883i 3.57784i −4.59437 3.33801i −5.23870 + 7.21045i −10.5862 3.43967i 5.88412 3.79172i 4.22583 3.07025i −3.80091 −16.2981 22.4324i
139.9 −0.875068 + 2.69318i 3.79026i −3.25142 2.36230i 2.82818 3.89266i 10.2079 + 3.31674i 6.36840 2.90576i 0.0434828 0.0315921i −5.36606 8.00880 + 11.0232i
139.10 −0.875068 + 2.69318i 3.79026i −3.25142 2.36230i −2.82818 + 3.89266i −10.2079 3.31674i −6.86011 + 1.39244i 0.0434828 0.0315921i −5.36606 −8.00880 11.0232i
139.11 −0.862122 + 2.65334i 5.31208i −3.06088 2.22386i −1.17908 + 1.62286i 14.0948 + 4.57966i 1.57310 + 6.82095i −0.488754 + 0.355101i −19.2182 −3.28949 4.52759i
139.12 −0.862122 + 2.65334i 5.31208i −3.06088 2.22386i 1.17908 1.62286i −14.0948 4.57966i 2.73659 + 6.44291i −0.488754 + 0.355101i −19.2182 3.28949 + 4.52759i
139.13 −0.695933 + 2.14186i 1.83433i −0.867179 0.630043i −0.161169 + 0.221830i 3.92888 + 1.27657i −1.23898 6.88948i −5.33494 + 3.87606i 5.63523 −0.362966 0.499580i
139.14 −0.695933 + 2.14186i 1.83433i −0.867179 0.630043i 0.161169 0.221830i −3.92888 1.27657i −3.04717 6.30196i −5.33494 + 3.87606i 5.63523 0.362966 + 0.499580i
139.15 −0.676052 + 2.08067i 3.43943i −0.636093 0.462148i −5.76773 + 7.93860i 7.15633 + 2.32523i −4.27053 5.54640i −5.68810 + 4.13265i −2.82965 −12.6184 17.3677i
139.16 −0.676052 + 2.08067i 3.43943i −0.636093 0.462148i 5.76773 7.93860i −7.15633 2.32523i 0.194841 6.99729i −5.68810 + 4.13265i −2.82965 12.6184 + 17.3677i
139.17 −0.566268 + 1.74279i 1.14923i 0.519396 + 0.377364i −0.885877 + 1.21931i 2.00287 + 0.650771i −4.74322 + 5.14800i −6.88182 + 4.99994i 7.67928 −1.62335 2.23436i
139.18 −0.566268 + 1.74279i 1.14923i 0.519396 + 0.377364i 0.885877 1.21931i −2.00287 0.650771i 6.86326 + 1.37682i −6.88182 + 4.99994i 7.67928 1.62335 + 2.23436i
139.19 −0.350708 + 1.07937i 2.32451i 2.19403 + 1.59406i 4.01634 5.52802i 2.50900 + 0.815222i 2.21612 + 6.63994i −6.16269 + 4.47746i 3.59668 4.55820 + 6.27383i
139.20 −0.350708 + 1.07937i 2.32451i 2.19403 + 1.59406i −4.01634 + 5.52802i −2.50900 0.815222i 2.10998 + 6.67443i −6.16269 + 4.47746i 3.59668 −4.55820 6.27383i
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.54 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
41.d even 5 1 inner
287.o odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.o.a 216
7.b odd 2 1 inner 287.3.o.a 216
41.d even 5 1 inner 287.3.o.a 216
287.o odd 10 1 inner 287.3.o.a 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.o.a 216 1.a even 1 1 trivial
287.3.o.a 216 7.b odd 2 1 inner
287.3.o.a 216 41.d even 5 1 inner
287.3.o.a 216 287.o odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(287, [\chi])$$.