Properties

Label 64.2.i.a
Level $64$
Weight $2$
Character orbit 64.i
Analytic conductor $0.511$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 64.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.511042572936\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 56q - 8q^{2} - 8q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 8q^{2} - 8q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 8q^{9} - 8q^{10} - 8q^{11} - 8q^{12} - 8q^{13} - 8q^{14} - 8q^{15} - 8q^{16} - 8q^{17} - 8q^{18} - 8q^{19} - 8q^{20} - 8q^{21} - 8q^{23} + 32q^{24} - 8q^{25} + 32q^{26} - 8q^{27} + 32q^{28} - 8q^{29} + 72q^{30} + 32q^{32} + 32q^{34} - 8q^{35} + 72q^{36} - 8q^{37} + 32q^{38} - 8q^{39} + 32q^{40} - 8q^{41} + 32q^{42} - 8q^{43} - 8q^{45} - 8q^{46} - 8q^{47} - 8q^{48} - 8q^{49} - 32q^{50} + 24q^{51} - 56q^{52} - 8q^{53} - 72q^{54} + 56q^{55} - 64q^{56} - 8q^{57} - 80q^{58} + 56q^{59} - 104q^{60} - 8q^{61} - 40q^{62} + 64q^{63} - 104q^{64} - 16q^{65} - 88q^{66} + 72q^{67} - 56q^{68} - 8q^{69} - 104q^{70} + 56q^{71} - 80q^{72} - 8q^{73} - 64q^{74} + 56q^{75} - 72q^{76} - 8q^{77} - 32q^{78} + 24q^{79} + 32q^{80} - 8q^{81} + 72q^{82} - 8q^{83} + 104q^{84} - 8q^{85} + 96q^{86} - 8q^{87} + 72q^{88} - 8q^{89} + 136q^{90} - 8q^{91} + 144q^{92} + 16q^{93} + 88q^{94} + 128q^{96} + 128q^{98} + 16q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.33595 + 0.463928i −1.31138 0.876237i 1.56954 1.23957i 3.52249 + 0.700667i 2.15846 + 0.562226i 1.02503 2.47464i −1.52176 + 2.38416i −0.196120 0.473476i −5.03094 + 0.698122i
5.2 −1.20609 + 0.738481i 2.51381 + 1.67967i 0.909293 1.78134i −2.28487 0.454489i −4.27227 0.169433i 0.303950 0.733799i 0.218802 + 2.81995i 2.34987 + 5.67309i 3.09139 1.13918i
5.3 −0.599818 1.28071i −2.03902 1.36243i −1.28044 + 1.53639i −1.53851 0.306028i −0.521836 + 3.42860i 1.01301 2.44562i 2.73569 + 0.718315i 1.15334 + 2.78440i 0.530891 + 2.15394i
5.4 0.0973444 + 1.41086i 0.306211 + 0.204603i −1.98105 + 0.274678i 1.42470 + 0.283390i −0.258859 + 0.451937i −0.666723 + 1.60961i −0.580376 2.76824i −1.09615 2.64634i −0.261137 + 2.03763i
5.5 0.406933 1.35440i 1.06920 + 0.714416i −1.66881 1.10230i −0.330507 0.0657419i 1.40270 1.15741i −0.739314 + 1.78486i −2.17206 + 1.81168i −0.515254 1.24393i −0.223535 + 0.420887i
5.6 1.19265 + 0.759993i 0.0799701 + 0.0534343i 0.844823 + 1.81281i −3.47403 0.691028i 0.0547666 + 0.124505i 1.22800 2.96465i −0.370144 + 2.80410i −1.14451 2.76309i −3.61812 3.46439i
5.7 1.36881 0.355465i −2.00147 1.33734i 1.74729 0.973128i 0.756852 + 0.150547i −3.21501 1.11911i −1.69148 + 4.08359i 2.04580 1.95313i 1.06935 + 2.58163i 1.08950 0.0629634i
13.1 −1.33595 0.463928i −1.31138 + 0.876237i 1.56954 + 1.23957i 3.52249 0.700667i 2.15846 0.562226i 1.02503 + 2.47464i −1.52176 2.38416i −0.196120 + 0.473476i −5.03094 0.698122i
13.2 −1.20609 0.738481i 2.51381 1.67967i 0.909293 + 1.78134i −2.28487 + 0.454489i −4.27227 + 0.169433i 0.303950 + 0.733799i 0.218802 2.81995i 2.34987 5.67309i 3.09139 + 1.13918i
13.3 −0.599818 + 1.28071i −2.03902 + 1.36243i −1.28044 1.53639i −1.53851 + 0.306028i −0.521836 3.42860i 1.01301 + 2.44562i 2.73569 0.718315i 1.15334 2.78440i 0.530891 2.15394i
13.4 0.0973444 1.41086i 0.306211 0.204603i −1.98105 0.274678i 1.42470 0.283390i −0.258859 0.451937i −0.666723 1.60961i −0.580376 + 2.76824i −1.09615 + 2.64634i −0.261137 2.03763i
13.5 0.406933 + 1.35440i 1.06920 0.714416i −1.66881 + 1.10230i −0.330507 + 0.0657419i 1.40270 + 1.15741i −0.739314 1.78486i −2.17206 1.81168i −0.515254 + 1.24393i −0.223535 0.420887i
13.6 1.19265 0.759993i 0.0799701 0.0534343i 0.844823 1.81281i −3.47403 + 0.691028i 0.0547666 0.124505i 1.22800 + 2.96465i −0.370144 2.80410i −1.14451 + 2.76309i −3.61812 + 3.46439i
13.7 1.36881 + 0.355465i −2.00147 + 1.33734i 1.74729 + 0.973128i 0.756852 0.150547i −3.21501 + 1.11911i −1.69148 4.08359i 2.04580 + 1.95313i 1.06935 2.58163i 1.08950 + 0.0629634i
21.1 −1.30780 0.538195i 0.344545 + 1.73215i 1.42069 + 1.40771i −2.21982 + 1.48324i 0.481636 2.45074i 2.90595 + 1.20368i −1.10036 2.60561i −0.109979 + 0.0455548i 3.70136 0.745083i
21.2 −1.27161 + 0.618873i −0.216111 1.08646i 1.23399 1.57393i 1.50133 1.00316i 0.947192 + 1.24781i 1.15320 + 0.477669i −0.595096 + 2.76512i 1.63794 0.678457i −1.28828 + 2.20476i
21.3 −0.887839 1.10079i −0.435353 2.18867i −0.423482 + 1.95465i 0.649649 0.434082i −2.02274 + 2.42242i −3.64486 1.50975i 2.52765 1.26925i −1.82909 + 0.757635i −1.05462 0.329733i
21.4 0.0941531 1.41108i 0.553854 + 2.78441i −1.98227 0.265714i 2.59756 1.73564i 3.98116 0.519369i −1.96508 0.813965i −0.561580 + 2.77212i −4.67456 + 1.93627i −2.20455 3.82878i
21.5 0.297937 + 1.38247i 0.123576 + 0.621259i −1.82247 + 0.823780i 0.660623 0.441414i −0.822057 + 0.355937i −0.860072 0.356253i −1.68183 2.27408i 2.40095 0.994505i 0.807067 + 0.781780i
21.6 1.19357 + 0.758552i −0.599600 3.01439i 0.849199 + 1.81076i −1.78465 + 1.19247i 1.57091 4.05270i 1.99271 + 0.825409i −0.359981 + 2.80543i −5.95540 + 2.46681i −3.03465 + 0.0695368i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.7
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 64.2.i.a 56
3.b odd 2 1 576.2.bd.a 56
4.b odd 2 1 256.2.i.a 56
8.b even 2 1 512.2.i.b 56
8.d odd 2 1 512.2.i.a 56
64.i even 16 1 inner 64.2.i.a 56
64.i even 16 1 512.2.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 1.a even 1 1 trivial
64.2.i.a 56 64.i even 16 1 inner
256.2.i.a 56 4.b odd 2 1
256.2.i.a 56 64.j odd 16 1
512.2.i.a 56 8.d odd 2 1
512.2.i.a 56 64.j odd 16 1
512.2.i.b 56 8.b even 2 1
512.2.i.b 56 64.i even 16 1
576.2.bd.a 56 3.b odd 2 1
576.2.bd.a 56 192.q odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(64, [\chi])\).