# Properties

 Label 256.2.i.a Level $256$ Weight $2$ Character orbit 256.i Analytic conductor $2.044$ Analytic rank $0$ Dimension $56$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 256.i (of order $$16$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.04417029174$$ 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −1.40926 2.10911i 0 −0.573974 2.88556i 0 0.410118 + 0.990113i 0 −1.31428 + 3.17295i 0
17.2 0 −1.25103 1.87230i 0 0.509835 + 2.56311i 0 1.78664 + 4.31333i 0 −0.792379 + 1.91297i 0
17.3 0 −0.443610 0.663909i 0 0.154331 + 0.775873i 0 −1.53949 3.71667i 0 0.904065 2.18261i 0
17.4 0 0.477374 + 0.714441i 0 0.0517508 + 0.260169i 0 0.515195 + 1.24379i 0 0.865510 2.08953i 0
17.5 0 0.572535 + 0.856859i 0 −0.690473 3.47124i 0 −0.983337 2.37399i 0 0.741639 1.79047i 0
17.6 0 0.894167 + 1.33822i 0 0.631428 + 3.17440i 0 0.127129 + 0.306917i 0 0.156763 0.378460i 0
17.7 0 1.77714 + 2.65968i 0 −0.159018 0.799435i 0 0.742008 + 1.79137i 0 −2.76763 + 6.68165i 0
49.1 0 −1.97142 + 0.392140i 0 −0.153107 0.229142i 0 0.843108 + 0.349227i 0 0.961093 0.398098i 0
49.2 0 −1.93660 + 0.385213i 0 −0.787711 1.17889i 0 2.16489 + 0.896725i 0 0.830380 0.343954i 0
49.3 0 −0.416408 + 0.0828287i 0 1.82421 + 2.73012i 0 −0.00395016 0.00163621i 0 −2.60510 + 1.07907i 0
49.4 0 −0.191980 + 0.0381873i 0 −0.967135 1.44742i 0 −4.53283 1.87756i 0 −2.73624 + 1.13339i 0
49.5 0 1.22190 0.243052i 0 0.884671 + 1.32400i 0 2.40727 + 0.997123i 0 −1.33766 + 0.554078i 0
49.6 0 2.23702 0.444970i 0 −2.33237 3.49064i 0 1.63661 + 0.677907i 0 2.03460 0.842759i 0
49.7 0 2.98137 0.593031i 0 0.914126 + 1.36809i 0 −2.65574 1.10004i 0 5.76522 2.38803i 0
81.1 0 −0.553854 + 2.78441i 0 2.59756 + 1.73564i 0 1.96508 0.813965i 0 −4.67456 1.93627i 0
81.2 0 −0.344545 + 1.73215i 0 −2.21982 1.48324i 0 −2.90595 + 1.20368i 0 −0.109979 0.0455548i 0
81.3 0 −0.152968 + 0.769021i 0 −2.78737 1.86246i 0 3.13672 1.29927i 0 2.20364 + 0.912779i 0
81.4 0 −0.123576 + 0.621259i 0 0.660623 + 0.441414i 0 0.860072 0.356253i 0 2.40095 + 0.994505i 0
81.5 0 0.216111 1.08646i 0 1.50133 + 1.00316i 0 −1.15320 + 0.477669i 0 1.63794 + 0.678457i 0
81.6 0 0.435353 2.18867i 0 0.649649 + 0.434082i 0 3.64486 1.50975i 0 −1.82909 0.757635i 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.7 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 256.2.i.a 56
4.b odd 2 1 64.2.i.a 56
8.b even 2 1 512.2.i.a 56
8.d odd 2 1 512.2.i.b 56
12.b even 2 1 576.2.bd.a 56
64.i even 16 1 inner 256.2.i.a 56
64.i even 16 1 512.2.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 4.b odd 2 1
64.2.i.a 56 64.j odd 16 1
256.2.i.a 56 1.a even 1 1 trivial
256.2.i.a 56 64.i even 16 1 inner
512.2.i.a 56 8.b even 2 1
512.2.i.a 56 64.i even 16 1
512.2.i.b 56 8.d odd 2 1
512.2.i.b 56 64.j odd 16 1
576.2.bd.a 56 12.b even 2 1
576.2.bd.a 56 192.s even 16 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(256, [\chi])$$.