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

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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:

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