# Properties

 Label 320.6.l.a Level 320 Weight 6 Character orbit 320.l Analytic conductor 51.323 Analytic rank 0 Dimension 80 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.3228223402$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 80) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 1208q^{11} + 1800q^{15} - 2360q^{19} + 7464q^{27} - 8144q^{29} + 21296q^{37} - 32072q^{43} + 88360q^{47} - 192080q^{49} + 5920q^{51} - 49456q^{53} - 44984q^{59} + 48080q^{61} - 158760q^{63} - 61160q^{67} - 22320q^{69} - 14896q^{77} - 177680q^{79} - 524880q^{81} + 329240q^{83} + 132400q^{85} - 364832q^{91} - 362352q^{93} - 288800q^{95} - 659000q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1 0 −21.2511 21.2511i 0 −17.6777 + 17.6777i 0 169.621i 0 660.222i 0
81.2 0 −21.1465 21.1465i 0 17.6777 17.6777i 0 100.160i 0 651.346i 0
81.3 0 −19.6483 19.6483i 0 −17.6777 + 17.6777i 0 152.043i 0 529.109i 0
81.4 0 −17.6894 17.6894i 0 17.6777 17.6777i 0 145.744i 0 382.828i 0
81.5 0 −16.3104 16.3104i 0 −17.6777 + 17.6777i 0 214.455i 0 289.056i 0
81.6 0 −16.2010 16.2010i 0 17.6777 17.6777i 0 159.660i 0 281.948i 0
81.7 0 −15.7717 15.7717i 0 −17.6777 + 17.6777i 0 14.9661i 0 254.492i 0
81.8 0 −14.9697 14.9697i 0 17.6777 17.6777i 0 66.6523i 0 205.184i 0
81.9 0 −11.6877 11.6877i 0 −17.6777 + 17.6777i 0 158.930i 0 30.2029i 0
81.10 0 −10.2463 10.2463i 0 −17.6777 + 17.6777i 0 107.195i 0 33.0270i 0
81.11 0 −9.88924 9.88924i 0 −17.6777 + 17.6777i 0 122.335i 0 47.4058i 0
81.12 0 −9.77316 9.77316i 0 17.6777 17.6777i 0 103.649i 0 51.9707i 0
81.13 0 −7.88400 7.88400i 0 17.6777 17.6777i 0 208.047i 0 118.685i 0
81.14 0 −6.55397 6.55397i 0 17.6777 17.6777i 0 188.280i 0 157.091i 0
81.15 0 −5.64986 5.64986i 0 −17.6777 + 17.6777i 0 143.077i 0 179.158i 0
81.16 0 −4.24643 4.24643i 0 17.6777 17.6777i 0 137.329i 0 206.936i 0
81.17 0 −3.86003 3.86003i 0 −17.6777 + 17.6777i 0 20.3706i 0 213.200i 0
81.18 0 −3.58258 3.58258i 0 17.6777 17.6777i 0 149.558i 0 217.330i 0
81.19 0 −2.82595 2.82595i 0 17.6777 17.6777i 0 197.416i 0 227.028i 0
81.20 0 −2.21237 2.21237i 0 −17.6777 + 17.6777i 0 21.9533i 0 233.211i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.40 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 320.6.l.a 80
4.b odd 2 1 80.6.l.a 80
16.e even 4 1 inner 320.6.l.a 80
16.f odd 4 1 80.6.l.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.l.a 80 4.b odd 2 1
80.6.l.a 80 16.f odd 4 1
320.6.l.a 80 1.a even 1 1 trivial
320.6.l.a 80 16.e even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database