Properties

 Label 512.2.i.a Level $512$ Weight $2$ Character orbit 512.i Analytic conductor $4.088$ Analytic rank $0$ Dimension $56$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.i (of order $$16$$, degree $$8$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.08834058349$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$7$$ over $$\Q(\zeta_{16})$$ Twist minimal: no (minimal twist has level 64) Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 8q^{3} + 8q^{5} + 8q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 8q^{3} + 8q^{5} + 8q^{7} - 8q^{9} - 8q^{11} + 8q^{13} + 8q^{15} - 8q^{17} - 8q^{19} + 8q^{21} + 8q^{23} - 8q^{25} - 8q^{27} + 8q^{29} - 8q^{35} + 8q^{37} + 8q^{39} - 8q^{41} - 8q^{43} + 8q^{45} + 8q^{47} - 8q^{49} + 24q^{51} + 8q^{53} - 56q^{55} - 8q^{57} + 56q^{59} + 8q^{61} - 64q^{63} - 16q^{65} + 72q^{67} + 8q^{69} - 56q^{71} - 8q^{73} + 56q^{75} + 8q^{77} - 24q^{79} - 8q^{81} - 8q^{83} + 8q^{85} + 8q^{87} - 8q^{89} - 8q^{91} - 16q^{93} + 16q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1 0 −2.03902 + 1.36243i 0 1.53851 0.306028i 0 −1.01301 2.44562i 0 1.15334 2.78440i 0
33.2 0 −2.00147 + 1.33734i 0 −0.756852 + 0.150547i 0 1.69148 + 4.08359i 0 1.06935 2.58163i 0
33.3 0 −1.31138 + 0.876237i 0 −3.52249 + 0.700667i 0 −1.02503 2.47464i 0 −0.196120 + 0.473476i 0
33.4 0 0.0799701 0.0534343i 0 3.47403 0.691028i 0 −1.22800 2.96465i 0 −1.14451 + 2.76309i 0
33.5 0 0.306211 0.204603i 0 −1.42470 + 0.283390i 0 0.666723 + 1.60961i 0 −1.09615 + 2.64634i 0
33.6 0 1.06920 0.714416i 0 0.330507 0.0657419i 0 0.739314 + 1.78486i 0 −0.515254 + 1.24393i 0
33.7 0 2.51381 1.67967i 0 2.28487 0.454489i 0 −0.303950 0.733799i 0 2.34987 5.67309i 0
97.1 0 −0.599600 3.01439i 0 1.78465 1.19247i 0 −1.99271 0.825409i 0 −5.95540 + 2.46681i 0
97.2 0 −0.435353 2.18867i 0 −0.649649 + 0.434082i 0 3.64486 + 1.50975i 0 −1.82909 + 0.757635i 0
97.3 0 −0.216111 1.08646i 0 −1.50133 + 1.00316i 0 −1.15320 0.477669i 0 1.63794 0.678457i 0
97.4 0 0.123576 + 0.621259i 0 −0.660623 + 0.441414i 0 0.860072 + 0.356253i 0 2.40095 0.994505i 0
97.5 0 0.152968 + 0.769021i 0 2.78737 1.86246i 0 3.13672 + 1.29927i 0 2.20364 0.912779i 0
97.6 0 0.344545 + 1.73215i 0 2.21982 1.48324i 0 −2.90595 1.20368i 0 −0.109979 + 0.0455548i 0
97.7 0 0.553854 + 2.78441i 0 −2.59756 + 1.73564i 0 1.96508 + 0.813965i 0 −4.67456 + 1.93627i 0
161.1 0 −2.98137 0.593031i 0 −0.914126 + 1.36809i 0 −2.65574 + 1.10004i 0 5.76522 + 2.38803i 0
161.2 0 −2.23702 0.444970i 0 2.33237 3.49064i 0 1.63661 0.677907i 0 2.03460 + 0.842759i 0
161.3 0 −1.22190 0.243052i 0 −0.884671 + 1.32400i 0 2.40727 0.997123i 0 −1.33766 0.554078i 0
161.4 0 0.191980 + 0.0381873i 0 0.967135 1.44742i 0 −4.53283 + 1.87756i 0 −2.73624 1.13339i 0
161.5 0 0.416408 + 0.0828287i 0 −1.82421 + 2.73012i 0 −0.00395016 + 0.00163621i 0 −2.60510 1.07907i 0
161.6 0 1.93660 + 0.385213i 0 0.787711 1.17889i 0 2.16489 0.896725i 0 0.830380 + 0.343954i 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 481.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.i.a 56
4.b odd 2 1 512.2.i.b 56
8.b even 2 1 256.2.i.a 56
8.d odd 2 1 64.2.i.a 56
24.f even 2 1 576.2.bd.a 56
64.i even 16 1 256.2.i.a 56
64.i even 16 1 inner 512.2.i.a 56
64.j odd 16 1 64.2.i.a 56
64.j odd 16 1 512.2.i.b 56
192.s even 16 1 576.2.bd.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.i.a 56 8.d odd 2 1
64.2.i.a 56 64.j odd 16 1
256.2.i.a 56 8.b even 2 1
256.2.i.a 56 64.i even 16 1
512.2.i.a 56 1.a even 1 1 trivial
512.2.i.a 56 64.i even 16 1 inner
512.2.i.b 56 4.b odd 2 1
512.2.i.b 56 64.j odd 16 1
576.2.bd.a 56 24.f even 2 1
576.2.bd.a 56 192.s even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$14\!\cdots\!80$$$$T_{3}^{21} +$$$$24\!\cdots\!72$$$$T_{3}^{20} +$$$$24\!\cdots\!08$$$$T_{3}^{19} +$$$$19\!\cdots\!36$$$$T_{3}^{18} +$$$$19\!\cdots\!24$$$$T_{3}^{17} + 959158230624 T_{3}^{16} + 748771137024 T_{3}^{15} + 228341444352 T_{3}^{14} - 373009462784 T_{3}^{13} + 704752233216 T_{3}^{12} -$$$$10\!\cdots\!12$$$$T_{3}^{11} + 982465612800 T_{3}^{10} - 897614395392 T_{3}^{9} + 759115497024 T_{3}^{8} - 543103289856 T_{3}^{7} + 324151793408 T_{3}^{6} - 146883157504 T_{3}^{5} + 45788524672 T_{3}^{4} - 9110427136 T_{3}^{3} + 1086851328 T_{3}^{2} - 71201280 T_{3} + 2064512$$">$$T_{3}^{56} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(512, [\chi])$$.