# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,2,Mod(5,64)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("64.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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: 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 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 $$S_{2}^{\mathrm{new}}(64, [\chi])$$.