Properties

 Label 287.3.t.a Level 287 Weight 3 Character orbit 287.t Analytic conductor 7.820 Analytic rank 0 Dimension 432 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.t (of order $$20$$, degree $$8$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$432$$ Relative dimension: $$54$$ over $$\Q(\zeta_{20})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$432q - 20q^{2} + 196q^{4} - 8q^{7} - 20q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$432q - 20q^{2} + 196q^{4} - 8q^{7} - 20q^{8} - 126q^{14} + 8q^{15} - 428q^{16} + 36q^{18} - 10q^{21} - 40q^{22} - 12q^{23} - 472q^{25} - 98q^{28} + 532q^{29} - 356q^{30} + 100q^{35} + 300q^{36} - 312q^{37} - 20q^{39} - 136q^{42} + 160q^{43} + 416q^{44} + 980q^{46} - 190q^{49} + 408q^{51} + 72q^{53} - 454q^{56} - 244q^{57} - 268q^{58} - 60q^{60} + 732q^{63} + 1164q^{64} + 624q^{65} + 328q^{67} - 1440q^{70} - 356q^{71} + 464q^{72} - 20q^{74} - 560q^{77} - 1944q^{78} - 216q^{79} - 2992q^{81} + 1390q^{84} - 52q^{85} - 172q^{86} - 380q^{88} + 228q^{92} + 588q^{93} - 24q^{95} - 228q^{98} + 2084q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1 −3.60896 1.17262i −3.31582 3.31582i 8.41351 + 6.11277i −1.82266 1.32424i 8.07846 + 15.8549i −4.25521 5.55816i −14.2742 19.6468i 12.9893i 5.02507 + 6.91642i
20.2 −3.60896 1.17262i 3.31582 + 3.31582i 8.41351 + 6.11277i 1.82266 + 1.32424i −8.07846 15.8549i −6.99780 0.175532i −14.2742 19.6468i 12.9893i −5.02507 6.91642i
20.3 −3.56829 1.15941i −0.388163 0.388163i 8.15243 + 5.92309i −4.84900 3.52300i 0.935040 + 1.83512i 1.52673 + 6.83148i −13.4017 18.4458i 8.69866i 13.2180 + 18.1931i
20.4 −3.56829 1.15941i 0.388163 + 0.388163i 8.15243 + 5.92309i 4.84900 + 3.52300i −0.935040 1.83512i 6.42417 2.78029i −13.4017 18.4458i 8.69866i −13.2180 18.1931i
20.5 −3.15113 1.02386i −3.91900 3.91900i 5.64524 + 4.10151i 4.45196 + 3.23454i 8.33674 + 16.3618i 4.30825 + 5.51716i −5.79947 7.98228i 21.7171i −10.7170 14.7506i
20.6 −3.15113 1.02386i 3.91900 + 3.91900i 5.64524 + 4.10151i −4.45196 3.23454i −8.33674 16.3618i 6.99580 + 0.242543i −5.79947 7.98228i 21.7171i 10.7170 + 14.7506i
20.7 −2.91771 0.948022i −1.59164 1.59164i 4.37823 + 3.18097i 2.45492 + 1.78360i 3.13503 + 6.15284i −4.48908 + 5.37104i −2.54581 3.50401i 3.93339i −5.47186 7.53136i
20.8 −2.91771 0.948022i 1.59164 + 1.59164i 4.37823 + 3.18097i −2.45492 1.78360i −3.13503 6.15284i 1.70665 6.78877i −2.54581 3.50401i 3.93339i 5.47186 + 7.53136i
20.9 −2.90133 0.942701i −1.41180 1.41180i 4.29299 + 3.11904i −4.40256 3.19865i 2.76519 + 5.42699i −3.67353 5.95862i −2.34258 3.22429i 5.01366i 9.75793 + 13.4306i
20.10 −2.90133 0.942701i 1.41180 + 1.41180i 4.29299 + 3.11904i 4.40256 + 3.19865i −2.76519 5.42699i −6.97987 + 0.530441i −2.34258 3.22429i 5.01366i −9.75793 13.4306i
20.11 −2.39823 0.779234i −2.09587 2.09587i 1.90826 + 1.38643i 7.33610 + 5.32999i 3.39322 + 6.65957i −0.631320 6.97147i 2.43267 + 3.34828i 0.214634i −13.4404 18.4991i
20.12 −2.39823 0.779234i 2.09587 + 2.09587i 1.90826 + 1.38643i −7.33610 5.32999i −3.39322 6.65957i −6.01112 + 3.58698i 2.43267 + 3.34828i 0.214634i 13.4404 + 18.4991i
20.13 −2.36022 0.766883i −3.15575 3.15575i 1.74648 + 1.26889i −6.16033 4.47574i 5.02818 + 9.86836i 6.99999 + 0.00883669i 2.68581 + 3.69670i 10.9175i 11.1074 + 15.2880i
20.14 −2.36022 0.766883i 3.15575 + 3.15575i 1.74648 + 1.26889i 6.16033 + 4.47574i −5.02818 9.86836i 4.12164 + 5.65792i 2.68581 + 3.69670i 10.9175i −11.1074 15.2880i
20.15 −1.95732 0.635971i −0.402457 0.402457i 0.190566 + 0.138454i −3.05686 2.22094i 0.531785 + 1.04369i 6.12162 + 3.39496i 4.55381 + 6.26778i 8.67606i 4.57079 + 6.29115i
20.16 −1.95732 0.635971i 0.402457 + 0.402457i 0.190566 + 0.138454i 3.05686 + 2.22094i −0.531785 1.04369i 6.34478 + 2.95699i 4.55381 + 6.26778i 8.67606i −4.57079 6.29115i
20.17 −1.62424 0.527747i −2.43233 2.43233i −0.876434 0.636766i 1.95820 + 1.42272i 2.66703 + 5.23435i 3.54135 6.03811i 5.10282 + 7.02344i 2.83249i −2.42975 3.34427i
20.18 −1.62424 0.527747i 2.43233 + 2.43233i −0.876434 0.636766i −1.95820 1.42272i −2.66703 5.23435i −2.80338 + 6.41413i 5.10282 + 7.02344i 2.83249i 2.42975 + 3.34427i
20.19 −1.56087 0.507158i −3.90043 3.90043i −1.05696 0.767924i −1.95861 1.42301i 4.10994 + 8.06621i −6.98802 + 0.409296i 5.11900 + 7.04570i 21.4267i 2.33544 + 3.21446i
20.20 −1.56087 0.507158i 3.90043 + 3.90043i −1.05696 0.767924i 1.95861 + 1.42301i −4.10994 8.06621i −3.77633 5.89401i 5.11900 + 7.04570i 21.4267i −2.33544 3.21446i
See next 80 embeddings (of 432 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 279.54 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.g even 20 1 inner
287.t odd 20 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.t.a 432
7.b odd 2 1 inner 287.3.t.a 432
41.g even 20 1 inner 287.3.t.a 432
287.t odd 20 1 inner 287.3.t.a 432

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.t.a 432 1.a even 1 1 trivial
287.3.t.a 432 7.b odd 2 1 inner
287.3.t.a 432 41.g even 20 1 inner
287.3.t.a 432 287.t odd 20 1 inner

Hecke kernels

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

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database