# Properties

 Label 64.8.e.a Level $64$ Weight $8$ Character orbit 64.e Analytic conductor $19.993$ Analytic rank $0$ Dimension $26$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(17,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("64.17");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.e (of order $$4$$, degree $$2$$, not 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: $$19.9926416310$$ Analytic rank: $$0$$ Dimension: $$26$$ Relative dimension: $$13$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$26 q + 2 q^{3} - 2 q^{5}+O(q^{10})$$ 26 * q + 2 * q^3 - 2 * q^5 $$\operatorname{Tr}(f)(q) =$$ $$26 q + 2 q^{3} - 2 q^{5} - 1202 q^{11} - 2 q^{13} + 27004 q^{15} - 4 q^{17} - 60582 q^{19} + 4372 q^{21} + 233672 q^{27} - 51690 q^{29} - 357488 q^{31} - 4 q^{33} + 252004 q^{35} + 415574 q^{37} - 569754 q^{43} + 151874 q^{45} + 2076464 q^{47} - 1647090 q^{49} - 2609508 q^{51} + 907814 q^{53} + 4865142 q^{59} + 2279886 q^{61} - 8295108 q^{63} - 1426892 q^{65} + 5564458 q^{67} - 4786076 q^{69} - 6212566 q^{75} + 7604308 q^{77} + 9598912 q^{79} - 5314414 q^{81} - 4531198 q^{83} + 7377748 q^{85} - 2587652 q^{91} - 14504144 q^{93} + 4900620 q^{95} - 4 q^{97} - 18815006 q^{99}+O(q^{100})$$ 26 * q + 2 * q^3 - 2 * q^5 - 1202 * q^11 - 2 * q^13 + 27004 * q^15 - 4 * q^17 - 60582 * q^19 + 4372 * q^21 + 233672 * q^27 - 51690 * q^29 - 357488 * q^31 - 4 * q^33 + 252004 * q^35 + 415574 * q^37 - 569754 * q^43 + 151874 * q^45 + 2076464 * q^47 - 1647090 * q^49 - 2609508 * q^51 + 907814 * q^53 + 4865142 * q^59 + 2279886 * q^61 - 8295108 * q^63 - 1426892 * q^65 + 5564458 * q^67 - 4786076 * q^69 - 6212566 * q^75 + 7604308 * q^77 + 9598912 * q^79 - 5314414 * q^81 - 4531198 * q^83 + 7377748 * q^85 - 2587652 * q^91 - 14504144 * q^93 + 4900620 * q^95 - 4 * q^97 - 18815006 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −61.7547 61.7547i 0 160.824 160.824i 0 1202.46i 0 5440.30i 0
17.2 0 −50.5872 50.5872i 0 −364.370 + 364.370i 0 182.346i 0 2931.14i 0
17.3 0 −42.2145 42.2145i 0 46.9348 46.9348i 0 1384.27i 0 1377.13i 0
17.4 0 −26.1364 26.1364i 0 −31.6175 + 31.6175i 0 444.381i 0 820.775i 0
17.5 0 −20.1662 20.1662i 0 269.345 269.345i 0 147.771i 0 1373.65i 0
17.6 0 −8.25573 8.25573i 0 63.0500 63.0500i 0 847.676i 0 2050.69i 0
17.7 0 −2.27414 2.27414i 0 −242.456 + 242.456i 0 1642.72i 0 2176.66i 0
17.8 0 12.2643 + 12.2643i 0 −210.432 + 210.432i 0 920.183i 0 1886.17i 0
17.9 0 21.1050 + 21.1050i 0 328.807 328.807i 0 874.718i 0 1296.15i 0
17.10 0 35.6625 + 35.6625i 0 50.2015 50.2015i 0 699.226i 0 356.628i 0
17.11 0 37.3394 + 37.3394i 0 −233.214 + 233.214i 0 241.506i 0 601.466i 0
17.12 0 49.4488 + 49.4488i 0 252.094 252.094i 0 1482.88i 0 2703.37i 0
17.13 0 56.5688 + 56.5688i 0 −90.1684 + 90.1684i 0 373.897i 0 4213.06i 0
49.1 0 −61.7547 + 61.7547i 0 160.824 + 160.824i 0 1202.46i 0 5440.30i 0
49.2 0 −50.5872 + 50.5872i 0 −364.370 364.370i 0 182.346i 0 2931.14i 0
49.3 0 −42.2145 + 42.2145i 0 46.9348 + 46.9348i 0 1384.27i 0 1377.13i 0
49.4 0 −26.1364 + 26.1364i 0 −31.6175 31.6175i 0 444.381i 0 820.775i 0
49.5 0 −20.1662 + 20.1662i 0 269.345 + 269.345i 0 147.771i 0 1373.65i 0
49.6 0 −8.25573 + 8.25573i 0 63.0500 + 63.0500i 0 847.676i 0 2050.69i 0
49.7 0 −2.27414 + 2.27414i 0 −242.456 242.456i 0 1642.72i 0 2176.66i 0
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.8.e.a 26
4.b odd 2 1 16.8.e.a 26
8.b even 2 1 128.8.e.a 26
8.d odd 2 1 128.8.e.b 26
16.e even 4 1 inner 64.8.e.a 26
16.e even 4 1 128.8.e.a 26
16.f odd 4 1 16.8.e.a 26
16.f odd 4 1 128.8.e.b 26

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.8.e.a 26 4.b odd 2 1
16.8.e.a 26 16.f odd 4 1
64.8.e.a 26 1.a even 1 1 trivial
64.8.e.a 26 16.e even 4 1 inner
128.8.e.a 26 8.b even 2 1
128.8.e.a 26 16.e even 4 1
128.8.e.b 26 8.d odd 2 1
128.8.e.b 26 16.f odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(64, [\chi])$$.