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Properties

Label 108.5.k.a
Level $108$
Weight $5$
Character orbit 108.k
Analytic conductor $11.164$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,5,Mod(5,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.k (of order \(18\), degree \(6\), 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: \(11.1639560131\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 0 −8.89497 − 1.37096i 0 −1.59659 − 4.38660i 0 12.9775 + 73.5989i 0 77.2410 + 24.3892i 0
5.2 0 −8.09115 − 3.94122i 0 14.0198 + 38.5191i 0 −13.2730 − 75.2747i 0 49.9335 + 63.7781i 0
5.3 0 −7.74246 + 4.58850i 0 −2.39469 − 6.57935i 0 −4.62159 − 26.2104i 0 38.8914 − 71.0525i 0
5.4 0 −5.53499 − 7.09676i 0 −14.0519 − 38.6073i 0 −5.57324 − 31.6074i 0 −19.7279 + 78.5609i 0
5.5 0 −2.33607 + 8.69153i 0 15.1037 + 41.4971i 0 8.53787 + 48.4207i 0 −70.0855 − 40.6081i 0
5.6 0 −1.23486 − 8.91488i 0 5.87553 + 16.1429i 0 8.12577 + 46.0835i 0 −77.9503 + 22.0172i 0
5.7 0 −0.126357 + 8.99911i 0 −6.31102 − 17.3394i 0 2.68521 + 15.2286i 0 −80.9681 − 2.27420i 0
5.8 0 4.78712 + 7.62125i 0 −3.22217 − 8.85284i 0 −16.2288 − 92.0381i 0 −35.1670 + 72.9677i 0
5.9 0 5.44219 − 7.16816i 0 4.60350 + 12.6480i 0 −6.68463 − 37.9104i 0 −21.7651 − 78.0210i 0
5.10 0 6.98131 − 5.67991i 0 −8.75895 − 24.0650i 0 −0.632293 − 3.58591i 0 16.4774 − 79.3063i 0
5.11 0 8.08622 + 3.95134i 0 −10.7879 − 29.6395i 0 15.2291 + 86.3688i 0 49.7738 + 63.9028i 0
5.12 0 8.73547 + 2.16600i 0 9.54164 + 26.2154i 0 −0.541932 − 3.07345i 0 71.6168 + 37.8421i 0
29.1 0 −8.99629 − 0.258464i 0 −25.3131 − 30.1670i 0 −27.6070 + 10.0481i 0 80.8664 + 4.65043i 0
29.2 0 −8.23149 + 3.63904i 0 18.1168 + 21.5908i 0 22.9581 − 8.35608i 0 54.5147 − 59.9095i 0
29.3 0 −6.90497 − 5.77247i 0 11.0592 + 13.1798i 0 7.77764 − 2.83083i 0 14.3573 + 79.7174i 0
29.4 0 −3.84126 + 8.13909i 0 0.0201223 + 0.0239808i 0 −60.3934 + 21.9814i 0 −51.4895 − 62.5287i 0
29.5 0 −2.24301 + 8.71602i 0 −14.1660 − 16.8824i 0 35.7790 − 13.0225i 0 −70.9378 − 39.1002i 0
29.6 0 −1.21427 − 8.91771i 0 −22.3567 − 26.6437i 0 80.4961 − 29.2982i 0 −78.0511 + 21.6570i 0
29.7 0 1.21143 − 8.91810i 0 1.79646 + 2.14093i 0 −63.3056 + 23.0413i 0 −78.0649 − 21.6073i 0
29.8 0 4.39288 + 7.85510i 0 6.32074 + 7.53276i 0 66.1128 − 24.0631i 0 −42.4052 + 69.0131i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.5.k.a ✓ 72
3.b odd 2 1 324.5.k.a 72
27.e even 9 1 324.5.k.a 72
27.f odd 18 1 inner 108.5.k.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.k.a ✓ 72 1.a even 1 1 trivial
108.5.k.a ✓ 72 27.f odd 18 1 inner
324.5.k.a 72 3.b odd 2 1
324.5.k.a 72 27.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(108, [\chi])\).