Properties

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

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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 −2.51381 + 1.67967i 0 2.28487 0.454489i 0 0.303950 + 0.733799i 0 2.34987 5.67309i 0
33.2 0 −1.06920 + 0.714416i 0 0.330507 0.0657419i 0 −0.739314 1.78486i 0 −0.515254 + 1.24393i 0
33.3 0 −0.306211 + 0.204603i 0 −1.42470 + 0.283390i 0 −0.666723 1.60961i 0 −1.09615 + 2.64634i 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 1.31138 0.876237i 0 −3.52249 + 0.700667i 0 1.02503 + 2.47464i 0 −0.196120 + 0.473476i 0
33.6 0 2.00147 1.33734i 0 −0.756852 + 0.150547i 0 −1.69148 4.08359i 0 1.06935 2.58163i 0
33.7 0 2.03902 1.36243i 0 1.53851 0.306028i 0 1.01301 + 2.44562i 0 1.15334 2.78440i 0
97.1 0 −0.553854 2.78441i 0 −2.59756 + 1.73564i 0 −1.96508 0.813965i 0 −4.67456 + 1.93627i 0
97.2 0 −0.344545 1.73215i 0 2.21982 1.48324i 0 2.90595 + 1.20368i 0 −0.109979 + 0.0455548i 0
97.3 0 −0.152968 0.769021i 0 2.78737 1.86246i 0 −3.13672 1.29927i 0 2.20364 0.912779i 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 0.216111 + 1.08646i 0 −1.50133 + 1.00316i 0 1.15320 + 0.477669i 0 1.63794 0.678457i 0
97.6 0 0.435353 + 2.18867i 0 −0.649649 + 0.434082i 0 −3.64486 1.50975i 0 −1.82909 + 0.757635i 0
97.7 0 0.599600 + 3.01439i 0 1.78465 1.19247i 0 1.99271 + 0.825409i 0 −5.95540 + 2.46681i 0
161.1 0 −1.97142 0.392140i 0 0.153107 0.229142i 0 −0.843108 + 0.349227i 0 0.961093 + 0.398098i 0
161.2 0 −1.93660 0.385213i 0 0.787711 1.17889i 0 −2.16489 + 0.896725i 0 0.830380 + 0.343954i 0
161.3 0 −0.416408 0.0828287i 0 −1.82421 + 2.73012i 0 0.00395016 0.00163621i 0 −2.60510 1.07907i 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 1.22190 + 0.243052i 0 −0.884671 + 1.32400i 0 −2.40727 + 0.997123i 0 −1.33766 0.554078i 0
161.6 0 2.23702 + 0.444970i 0 2.33237 3.49064i 0 −1.63661 + 0.677907i 0 2.03460 + 0.842759i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.7
Significant digits:
Format:

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.b 56
4.b odd 2 1 512.2.i.a 56
8.b even 2 1 64.2.i.a 56
8.d odd 2 1 256.2.i.a 56
24.h odd 2 1 576.2.bd.a 56
64.i even 16 1 64.2.i.a 56
64.i even 16 1 inner 512.2.i.b 56
64.j odd 16 1 256.2.i.a 56
64.j odd 16 1 512.2.i.a 56
192.q odd 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.b even 2 1
64.2.i.a 56 64.i even 16 1
256.2.i.a 56 8.d odd 2 1
256.2.i.a 56 64.j odd 16 1
512.2.i.a 56 4.b odd 2 1
512.2.i.a 56 64.j odd 16 1
512.2.i.b 56 1.a even 1 1 trivial
512.2.i.b 56 64.i even 16 1 inner
576.2.bd.a 56 24.h odd 2 1
576.2.bd.a 56 192.q odd 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 \(S_{2}^{\mathrm{new}}(512, [\chi])\).