# Properties

 Label 287.3.bd.a Level 287 Weight 3 Character orbit 287.bd Analytic conductor 7.820 Analytic rank 0 Dimension 864 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.bd (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$864$$ Relative dimension: $$54$$ over $$\Q(\zeta_{60})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$864q - 10q^{2} - 24q^{3} - 214q^{4} - 30q^{5} - 16q^{7} - 40q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$864q - 10q^{2} - 24q^{3} - 214q^{4} - 30q^{5} - 16q^{7} - 40q^{8} - 18q^{10} - 186q^{14} - 56q^{15} + 362q^{16} - 78q^{17} - 54q^{18} + 48q^{19} - 20q^{21} + 40q^{22} - 6q^{23} - 138q^{24} + 454q^{25} - 66q^{26} + 74q^{28} - 640q^{29} - 22q^{30} + 54q^{31} - 180q^{33} - 142q^{35} - 360q^{36} - 156q^{37} - 6q^{38} - 10q^{39} - 300q^{40} - 200q^{42} + 320q^{43} + 112q^{44} - 210q^{45} + 490q^{46} + 252q^{47} + 160q^{49} + 168q^{51} + 276q^{52} + 234q^{53} - 1164q^{54} - 110q^{56} - 656q^{57} + 106q^{58} + 378q^{59} - 486q^{60} - 30q^{61} - 480q^{63} + 720q^{64} + 42q^{65} + 2442q^{66} + 284q^{67} - 2058q^{68} + 642q^{70} + 524q^{71} + 82q^{72} - 10q^{74} - 1512q^{75} - 640q^{77} + 1488q^{78} - 18q^{79} - 30q^{80} + 2608q^{81} + 672q^{82} - 1420q^{84} - 44q^{85} + 202q^{86} - 30q^{87} - 742q^{88} + 1314q^{89} + 492q^{92} - 768q^{93} - 3666q^{94} - 288q^{95} + 6492q^{96} - 690q^{98} - 1700q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −3.86332 + 0.406051i 0.385359 + 0.103257i 10.8478 2.30576i 5.36436 + 5.95772i −1.53069 0.242438i 0.114923 6.99906i −26.1942 + 8.51102i −7.65639 4.42042i −23.1434 20.8384i
5.2 −3.78322 + 0.397633i −2.36963 0.634941i 10.2421 2.17702i −4.89261 5.43380i 9.21731 + 1.45988i −4.50728 + 5.35578i −23.4109 + 7.60666i −2.58223 1.49085i 20.6705 + 18.6118i
5.3 −3.67223 + 0.385966i 3.60494 + 0.965940i 9.42368 2.00307i −3.30438 3.66989i −13.6110 2.15577i 6.98845 0.401949i −19.7858 + 6.42881i 4.26831 + 2.46431i 13.5509 + 12.2013i
5.4 −3.52452 + 0.370441i −5.59949 1.50038i 8.37239 1.77961i 5.79187 + 6.43253i 20.2913 + 3.21382i −3.57911 + 6.01581i −15.3675 + 4.99319i 21.3089 + 12.3027i −22.7964 20.5260i
5.5 −3.51461 + 0.369401i 3.58086 + 0.959489i 8.30344 1.76495i −1.51949 1.68757i −12.9398 2.04946i −6.56906 2.41814i −15.0874 + 4.90219i 4.10772 + 2.37160i 5.96381 + 5.36983i
5.6 −3.25516 + 0.342131i −2.98484 0.799785i 6.56644 1.39574i −1.83918 2.04262i 9.98976 + 1.58222i −2.67170 6.47009i −8.44572 + 2.74418i 0.475370 + 0.274455i 6.68568 + 6.01981i
5.7 −3.25016 + 0.341606i −3.96280 1.06183i 6.53427 1.38890i −1.12109 1.24510i 13.2425 + 2.09740i 6.77770 1.75009i −8.33052 + 2.70675i 6.78206 + 3.91562i 4.06906 + 3.66380i
5.8 −3.15003 + 0.331082i 5.01583 + 1.34399i 5.90050 1.25419i 3.23952 + 3.59786i −16.2450 2.57296i 4.02106 + 5.72984i −6.12207 + 1.98918i 15.5580 + 8.98244i −11.3958 10.2608i
5.9 −3.11963 + 0.327886i 1.13607 + 0.304409i 5.71200 1.21412i 1.95582 + 2.17215i −3.64393 0.577142i −4.33634 + 5.49510i −5.48808 + 1.78318i −6.59624 3.80834i −6.81364 6.13503i
5.10 −2.89312 + 0.304080i −0.510002 0.136655i 4.36511 0.927833i 3.45934 + 3.84199i 1.51705 + 0.240278i 6.80668 + 1.63372i −1.27995 + 0.415880i −7.55280 4.36061i −11.1766 10.0634i
5.11 −2.46927 + 0.259531i 0.688721 + 0.184542i 2.11735 0.450057i −5.13570 5.70377i −1.74853 0.276940i 2.59590 + 6.50087i 4.33389 1.40817i −7.35395 4.24580i 14.1617 + 12.7513i
5.12 −2.38273 + 0.250435i 1.40604 + 0.376749i 1.70209 0.361791i −4.10188 4.55560i −3.44457 0.545567i 3.31786 6.16375i 5.14935 1.67313i −5.95921 3.44055i 10.9146 + 9.82752i
5.13 −2.24269 + 0.235717i 4.18226 + 1.12063i 1.06153 0.225634i 5.21001 + 5.78631i −9.64368 1.52741i −2.17446 6.65370i 6.25122 2.03114i 8.44125 + 4.87356i −13.0484 11.7488i
5.14 −2.22184 + 0.233525i −2.69284 0.721545i 0.969471 0.206067i 2.30180 + 2.55641i 6.15158 + 0.974314i −6.38644 2.86590i 6.39307 2.07723i −1.06346 0.613988i −5.71124 5.14242i
5.15 −2.19003 + 0.230182i 5.08333 + 1.36208i 0.830673 0.176565i −3.40707 3.78394i −11.4462 1.81290i −6.83271 1.52119i 6.59871 2.14405i 16.1908 + 9.34777i 8.33260 + 7.50271i
5.16 −1.90739 + 0.200474i 1.60329 + 0.429599i −0.314656 + 0.0668823i 1.19878 + 1.33139i −3.14421 0.497994i 4.18213 5.61335i 7.88286 2.56130i −5.40825 3.12246i −2.55345 2.29914i
5.17 −1.84676 + 0.194102i −4.62713 1.23984i −0.539743 + 0.114726i −4.15770 4.61759i 8.78585 + 1.39154i 5.25002 + 4.63004i 8.03870 2.61193i 12.0789 + 6.97376i 8.57456 + 7.72057i
5.18 −1.80705 + 0.189928i −2.98183 0.798979i −0.683242 + 0.145228i 0.710415 + 0.788996i 5.54006 + 0.877459i −2.34483 + 6.59559i 8.11935 2.63814i 0.458720 + 0.264842i −1.43361 1.29082i
5.19 −1.32950 + 0.139736i −1.23979 0.332200i −2.16455 + 0.460089i −5.09510 5.65868i 1.69472 + 0.268417i −6.54965 2.47023i 7.89905 2.56656i −6.36751 3.67628i 7.56465 + 6.81124i
5.20 −1.22238 + 0.128477i −2.41703 0.647641i −2.43489 + 0.517552i 6.07573 + 6.74778i 3.03773 + 0.481129i 6.66096 + 2.15212i 7.58568 2.46474i −2.37164 1.36927i −8.29377 7.46774i
See next 80 embeddings (of 864 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 285.54 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.d odd 6 1 inner
41.g even 20 1 inner
287.bd odd 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.bd.a 864
7.d odd 6 1 inner 287.3.bd.a 864
41.g even 20 1 inner 287.3.bd.a 864
287.bd odd 60 1 inner 287.3.bd.a 864

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.bd.a 864 1.a even 1 1 trivial
287.3.bd.a 864 7.d odd 6 1 inner
287.3.bd.a 864 41.g even 20 1 inner
287.3.bd.a 864 287.bd odd 60 1 inner

## Hecke kernels

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