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Properties

Label 108.6.i.a
Level $108$
Weight $6$
Character orbit 108.i
Analytic conductor $17.321$
Analytic rank $0$
Dimension $90$
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] = [108,6,Mod(13,108)]
 
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, 8]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.13");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 108.i (of order \(9\), 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: \(17.3214525398\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −15.3241 + 2.85854i 0 −63.0232 − 52.8828i 0 −226.162 − 82.3161i 0 226.658 − 87.6091i 0
13.2 0 −14.7517 − 5.03867i 0 8.41214 + 7.05862i 0 −17.8438 − 6.49461i 0 192.224 + 148.658i 0
13.3 0 −14.2968 + 6.21296i 0 82.0644 + 68.8602i 0 4.01162 + 1.46011i 0 165.798 − 177.651i 0
13.4 0 −12.9624 + 8.65895i 0 −29.6003 − 24.8376i 0 188.011 + 68.4304i 0 93.0451 − 224.481i 0
13.5 0 −11.5186 − 10.5034i 0 0.194482 + 0.163190i 0 83.4487 + 30.3729i 0 22.3564 + 241.969i 0
13.6 0 −4.93759 + 14.7858i 0 −1.72634 − 1.44857i 0 22.1444 + 8.05989i 0 −194.240 − 146.012i 0
13.7 0 −2.03335 − 15.4553i 0 58.1486 + 48.7924i 0 −131.626 − 47.9080i 0 −234.731 + 62.8520i 0
13.8 0 −1.96043 + 15.4647i 0 25.7256 + 21.5863i 0 −179.693 − 65.4027i 0 −235.313 − 60.6348i 0
13.9 0 0.571582 − 15.5780i 0 −56.0600 − 47.0399i 0 −23.6165 − 8.59570i 0 −242.347 − 17.8082i 0
13.10 0 5.03917 + 14.7515i 0 −66.1309 − 55.4904i 0 74.2607 + 27.0287i 0 −192.214 + 148.671i 0
13.11 0 9.06551 − 12.6813i 0 40.6723 + 34.1282i 0 234.829 + 85.4708i 0 −78.6332 − 229.926i 0
13.12 0 11.6047 + 10.4082i 0 50.4250 + 42.3116i 0 105.429 + 38.3730i 0 26.3374 + 241.569i 0
13.13 0 13.5505 − 7.70608i 0 23.4643 + 19.6889i 0 −110.864 − 40.3512i 0 124.233 − 208.843i 0
13.14 0 14.3467 + 6.09679i 0 −19.6970 − 16.5278i 0 −125.630 − 45.7255i 0 168.658 + 174.938i 0
13.15 0 15.0066 − 4.21923i 0 −56.3178 − 47.2563i 0 103.300 + 37.5982i 0 207.396 − 126.633i 0
25.1 0 −15.3241 − 2.85854i 0 −63.0232 + 52.8828i 0 −226.162 + 82.3161i 0 226.658 + 87.6091i 0
25.2 0 −14.7517 + 5.03867i 0 8.41214 − 7.05862i 0 −17.8438 + 6.49461i 0 192.224 − 148.658i 0
25.3 0 −14.2968 − 6.21296i 0 82.0644 − 68.8602i 0 4.01162 − 1.46011i 0 165.798 + 177.651i 0
25.4 0 −12.9624 − 8.65895i 0 −29.6003 + 24.8376i 0 188.011 − 68.4304i 0 93.0451 + 224.481i 0
25.5 0 −11.5186 + 10.5034i 0 0.194482 − 0.163190i 0 83.4487 − 30.3729i 0 22.3564 − 241.969i 0
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.15
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 108.6.i.a ✓ 90
3.b odd 2 1 324.6.i.a 90
27.e even 9 1 inner 108.6.i.a ✓ 90
27.f odd 18 1 324.6.i.a 90
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.i.a ✓ 90 1.a even 1 1 trivial
108.6.i.a ✓ 90 27.e even 9 1 inner
324.6.i.a 90 3.b odd 2 1
324.6.i.a 90 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(108, [\chi])\).