Properties

Label 108.4.i.a
Level 108
Weight 4
Character orbit 108.i
Analytic conductor 6.372
Analytic rank 0
Dimension 54
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.37220628062\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54q + 12q^{5} - 48q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 54q + 12q^{5} - 48q^{9} - 87q^{11} + 234q^{15} + 204q^{17} - 12q^{21} + 96q^{23} - 216q^{25} + 27q^{27} + 318q^{29} - 54q^{31} + 63q^{33} + 6q^{35} + 66q^{39} + 867q^{41} - 513q^{43} - 306q^{45} - 1548q^{47} + 594q^{49} - 1368q^{51} - 1068q^{53} - 1269q^{57} - 1218q^{59} - 54q^{61} + 30q^{63} + 96q^{65} - 2997q^{67} + 1476q^{69} - 120q^{71} - 216q^{73} + 732q^{75} + 3480q^{77} + 2808q^{79} + 3348q^{81} + 4464q^{83} + 2160q^{85} + 4824q^{87} + 4029q^{89} + 270q^{91} + 1164q^{93} - 1650q^{95} - 3483q^{97} - 5076q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −4.52704 + 2.55067i 0 −1.73722 1.45770i 0 −5.32680 1.93880i 0 13.9881 23.0940i 0
13.2 0 −4.40571 2.75494i 0 8.18465 + 6.86774i 0 −24.4722 8.90714i 0 11.8206 + 24.2749i 0
13.3 0 −4.40211 2.76069i 0 −14.9393 12.5356i 0 23.5233 + 8.56177i 0 11.7572 + 24.3057i 0
13.4 0 −0.716497 5.14652i 0 7.52940 + 6.31792i 0 24.9076 + 9.06562i 0 −25.9733 + 7.37493i 0
13.5 0 −0.173032 + 5.19327i 0 6.54095 + 5.48851i 0 11.9650 + 4.35489i 0 −26.9401 1.79720i 0
13.6 0 2.34095 4.63896i 0 −5.19451 4.35871i 0 −17.8009 6.47901i 0 −16.0399 21.7191i 0
13.7 0 2.66172 + 4.46265i 0 −12.8040 10.7439i 0 −20.8033 7.57178i 0 −12.8305 + 23.7566i 0
13.8 0 5.14478 + 0.728858i 0 −5.18194 4.34817i 0 16.8240 + 6.12342i 0 25.9375 + 7.49962i 0
13.9 0 5.19029 0.246754i 0 15.7004 + 13.1742i 0 −8.81657 3.20897i 0 26.8782 2.56145i 0
25.1 0 −4.52704 2.55067i 0 −1.73722 + 1.45770i 0 −5.32680 + 1.93880i 0 13.9881 + 23.0940i 0
25.2 0 −4.40571 + 2.75494i 0 8.18465 6.86774i 0 −24.4722 + 8.90714i 0 11.8206 24.2749i 0
25.3 0 −4.40211 + 2.76069i 0 −14.9393 + 12.5356i 0 23.5233 8.56177i 0 11.7572 24.3057i 0
25.4 0 −0.716497 + 5.14652i 0 7.52940 6.31792i 0 24.9076 9.06562i 0 −25.9733 7.37493i 0
25.5 0 −0.173032 5.19327i 0 6.54095 5.48851i 0 11.9650 4.35489i 0 −26.9401 + 1.79720i 0
25.6 0 2.34095 + 4.63896i 0 −5.19451 + 4.35871i 0 −17.8009 + 6.47901i 0 −16.0399 + 21.7191i 0
25.7 0 2.66172 4.46265i 0 −12.8040 + 10.7439i 0 −20.8033 + 7.57178i 0 −12.8305 23.7566i 0
25.8 0 5.14478 0.728858i 0 −5.18194 + 4.34817i 0 16.8240 6.12342i 0 25.9375 7.49962i 0
25.9 0 5.19029 + 0.246754i 0 15.7004 13.1742i 0 −8.81657 + 3.20897i 0 26.8782 + 2.56145i 0
49.1 0 −5.16735 + 0.546343i 0 5.21695 1.89881i 0 1.58868 + 9.00985i 0 26.4030 5.64629i 0
49.2 0 −4.14053 + 3.13943i 0 −4.27250 + 1.55506i 0 −4.69568 26.6305i 0 7.28791 25.9978i 0
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.i.a 54
3.b odd 2 1 324.4.i.a 54
27.e even 9 1 inner 108.4.i.a 54
27.f odd 18 1 324.4.i.a 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.i.a 54 1.a even 1 1 trivial
108.4.i.a 54 27.e even 9 1 inner
324.4.i.a 54 3.b odd 2 1
324.4.i.a 54 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database