# Properties

 Label 287.3.g.a Level 287 Weight 3 Character orbit 287.g Analytic conductor 7.820 Analytic rank 0 Dimension 108 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$108$$ Relative dimension: $$54$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$108q - 216q^{4} - 2q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$108q - 216q^{4} - 2q^{7} - 20q^{11} - 4q^{14} - 28q^{15} + 408q^{16} + 24q^{18} + 20q^{22} - 8q^{23} + 452q^{25} - 62q^{28} + 168q^{29} - 64q^{30} - 10q^{35} - 208q^{37} - 44q^{42} + 44q^{44} + 272q^{51} - 92q^{53} + 24q^{56} - 256q^{57} + 248q^{58} - 440q^{60} - 252q^{63} - 1104q^{64} - 644q^{65} - 348q^{67} - 30q^{70} - 564q^{71} + 796q^{72} + 1924q^{78} + 196q^{79} - 468q^{81} + 32q^{85} + 152q^{86} + 840q^{88} - 428q^{92} + 32q^{93} + 4q^{95} + 108q^{98} - 484q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
132.1 3.88864i −1.07444 + 1.07444i −11.1215 6.46442 4.17812 + 4.17812i −6.77652 + 1.75466i 27.6932i 6.69115i 25.1378i
132.2 3.88864i 1.07444 1.07444i −11.1215 −6.46442 −4.17812 4.17812i 1.75466 6.77652i 27.6932i 6.69115i 25.1378i
132.3 3.66303i −3.03803 + 3.03803i −9.41780 −7.09262 11.1284 + 11.1284i 1.44819 + 6.84856i 19.8456i 9.45930i 25.9805i
132.4 3.66303i 3.03803 3.03803i −9.41780 7.09262 −11.1284 11.1284i 6.84856 + 1.44819i 19.8456i 9.45930i 25.9805i
132.5 3.39532i −3.29534 + 3.29534i −7.52821 3.85237 11.1888 + 11.1888i 3.61480 5.99443i 11.9794i 12.7186i 13.0800i
132.6 3.39532i 3.29534 3.29534i −7.52821 −3.85237 −11.1888 11.1888i −5.99443 + 3.61480i 11.9794i 12.7186i 13.0800i
132.7 3.19098i −0.547162 + 0.547162i −6.18236 3.96088 1.74598 + 1.74598i 6.79862 + 1.66698i 6.96386i 8.40123i 12.6391i
132.8 3.19098i 0.547162 0.547162i −6.18236 −3.96088 −1.74598 1.74598i 1.66698 + 6.79862i 6.96386i 8.40123i 12.6391i
132.9 2.69918i −0.291651 + 0.291651i −3.28560 −0.108748 0.787221 + 0.787221i −6.97500 + 0.591083i 1.92831i 8.82988i 0.293532i
132.10 2.69918i 0.291651 0.291651i −3.28560 0.108748 −0.787221 0.787221i 0.591083 6.97500i 1.92831i 8.82988i 0.293532i
132.11 2.66341i −2.47686 + 2.47686i −3.09375 −7.32409 6.59690 + 6.59690i −3.35766 6.14216i 2.41372i 3.26970i 19.5070i
132.12 2.66341i 2.47686 2.47686i −3.09375 7.32409 −6.59690 6.59690i −6.14216 3.35766i 2.41372i 3.26970i 19.5070i
132.13 2.64775i −3.79352 + 3.79352i −3.01059 1.69265 10.0443 + 10.0443i −5.12305 + 4.77016i 2.61971i 19.7816i 4.48173i
132.14 2.64775i 3.79352 3.79352i −3.01059 −1.69265 −10.0443 10.0443i 4.77016 5.12305i 2.61971i 19.7816i 4.48173i
132.15 1.73155i −2.85212 + 2.85212i 1.00173 4.47811 4.93859 + 4.93859i 4.16852 + 5.62348i 8.66075i 7.26914i 7.75408i
132.16 1.73155i 2.85212 2.85212i 1.00173 −4.47811 −4.93859 4.93859i 5.62348 + 4.16852i 8.66075i 7.26914i 7.75408i
132.17 1.66337i −2.23909 + 2.23909i 1.23319 −6.64247 3.72444 + 3.72444i 6.99925 0.102623i 8.70475i 1.02704i 11.0489i
132.18 1.66337i 2.23909 2.23909i 1.23319 6.64247 −3.72444 3.72444i −0.102623 + 6.99925i 8.70475i 1.02704i 11.0489i
132.19 1.34555i −2.45356 + 2.45356i 2.18949 9.16655 3.30139 + 3.30139i −4.09097 5.68014i 8.32828i 3.03988i 12.3341i
132.20 1.34555i 2.45356 2.45356i 2.18949 −9.16655 −3.30139 3.30139i −5.68014 4.09097i 8.32828i 3.03988i 12.3341i
See next 80 embeddings (of 108 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 237.54 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.c even 4 1 inner
287.g odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.g.a 108
7.b odd 2 1 inner 287.3.g.a 108
41.c even 4 1 inner 287.3.g.a 108
287.g odd 4 1 inner 287.3.g.a 108

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.g.a 108 1.a even 1 1 trivial
287.3.g.a 108 7.b odd 2 1 inner
287.3.g.a 108 41.c even 4 1 inner
287.3.g.a 108 287.g odd 4 1 inner

## Hecke kernels

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