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Properties

Label 108.4.h.b
Level $108$
Weight $4$
Character orbit 108.h
Analytic conductor $6.372$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,4,Mod(35,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.35");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.h (of order \(6\), degree \(2\), not 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: \(6.37220628062\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{4} + 72 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{4} + 72 q^{5} + 96 q^{10} - 216 q^{13} + 36 q^{14} - 72 q^{16} + 540 q^{20} - 192 q^{22} + 252 q^{25} - 672 q^{28} - 576 q^{29} - 360 q^{32} - 660 q^{34} + 1248 q^{37} + 144 q^{38} + 636 q^{40} - 1116 q^{41} + 960 q^{46} + 348 q^{49} + 648 q^{50} + 132 q^{52} + 1692 q^{56} + 516 q^{58} - 264 q^{61} + 960 q^{64} + 2592 q^{65} - 5688 q^{68} + 564 q^{70} - 4776 q^{73} - 5652 q^{74} - 600 q^{76} - 648 q^{77} - 4104 q^{82} + 720 q^{85} + 9540 q^{86} + 1956 q^{88} + 7416 q^{92} - 1188 q^{94} + 588 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −2.71307 + 0.799546i 0 6.72145 − 4.33844i 14.2911 − 8.25096i 0 −19.2620 − 11.1209i −14.7670 + 17.1446i 0 −32.1756 + 33.8118i
35.2 −2.66543 − 0.946295i 0 6.20905 + 5.04457i −2.08666 + 1.20474i 0 2.30362 + 1.33000i −11.7761 − 19.3216i 0 6.70190 − 1.23654i
35.3 −2.15223 − 1.83518i 0 1.26420 + 7.89948i −2.08666 + 1.20474i 0 −2.30362 − 1.33000i 11.7761 − 19.3216i 0 6.70190 + 1.23654i
35.4 −1.44536 + 2.43124i 0 −3.82189 − 7.02803i −14.6499 + 8.45813i 0 3.08966 + 1.78382i 22.6108 + 0.866066i 0 0.610574 − 47.8425i
35.5 −0.823719 + 2.70582i 0 −6.64298 − 4.45768i 4.71466 − 2.72201i 0 20.9358 + 12.0873i 17.5336 − 14.3029i 0 3.48173 + 14.9992i
35.6 −0.664105 − 2.74936i 0 −7.11793 + 3.65173i 14.2911 − 8.25096i 0 19.2620 + 11.1209i 14.7670 + 17.1446i 0 −32.1756 − 33.8118i
35.7 0.157323 + 2.82405i 0 −7.95050 + 0.888573i 1.23846 − 0.715028i 0 −23.8818 − 13.7882i −3.76017 − 22.3128i 0 2.21411 + 3.38499i
35.8 1.38284 − 2.46734i 0 −4.17551 − 6.82387i −14.6499 + 8.45813i 0 −3.08966 − 1.78382i −22.6108 + 0.866066i 0 0.610574 + 47.8425i
35.9 1.65391 + 2.29447i 0 −2.52915 + 7.58969i 14.4924 − 8.36717i 0 16.7175 + 9.65186i −21.5973 + 6.74964i 0 43.1673 + 19.4137i
35.10 1.93145 − 2.06627i 0 −0.538974 − 7.98182i 4.71466 − 2.72201i 0 −20.9358 − 12.0873i −17.5336 − 14.3029i 0 3.48173 − 14.9992i
35.11 2.52436 − 1.27578i 0 4.74478 − 6.44105i 1.23846 − 0.715028i 0 23.8818 + 13.7882i 3.76017 − 22.3128i 0 2.21411 − 3.38499i
35.12 2.81402 + 0.285097i 0 7.83744 + 1.60454i 14.4924 − 8.36717i 0 −16.7175 − 9.65186i 21.5973 + 6.74964i 0 43.1673 − 19.4137i
71.1 −2.71307 − 0.799546i 0 6.72145 + 4.33844i 14.2911 + 8.25096i 0 −19.2620 + 11.1209i −14.7670 − 17.1446i 0 −32.1756 − 33.8118i
71.2 −2.66543 + 0.946295i 0 6.20905 − 5.04457i −2.08666 − 1.20474i 0 2.30362 − 1.33000i −11.7761 + 19.3216i 0 6.70190 + 1.23654i
71.3 −2.15223 + 1.83518i 0 1.26420 − 7.89948i −2.08666 − 1.20474i 0 −2.30362 + 1.33000i 11.7761 + 19.3216i 0 6.70190 − 1.23654i
71.4 −1.44536 − 2.43124i 0 −3.82189 + 7.02803i −14.6499 − 8.45813i 0 3.08966 − 1.78382i 22.6108 − 0.866066i 0 0.610574 + 47.8425i
71.5 −0.823719 − 2.70582i 0 −6.64298 + 4.45768i 4.71466 + 2.72201i 0 20.9358 − 12.0873i 17.5336 + 14.3029i 0 3.48173 − 14.9992i
71.6 −0.664105 + 2.74936i 0 −7.11793 − 3.65173i 14.2911 + 8.25096i 0 19.2620 − 11.1209i 14.7670 − 17.1446i 0 −32.1756 + 33.8118i
71.7 0.157323 − 2.82405i 0 −7.95050 − 0.888573i 1.23846 + 0.715028i 0 −23.8818 + 13.7882i −3.76017 + 22.3128i 0 2.21411 − 3.38499i
71.8 1.38284 + 2.46734i 0 −4.17551 + 6.82387i −14.6499 − 8.45813i 0 −3.08966 + 1.78382i −22.6108 − 0.866066i 0 0.610574 − 47.8425i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.h.b 24
3.b odd 2 1 36.4.h.b ✓ 24
4.b odd 2 1 inner 108.4.h.b 24
9.c even 3 1 36.4.h.b ✓ 24
9.c even 3 1 324.4.b.c 24
9.d odd 6 1 inner 108.4.h.b 24
9.d odd 6 1 324.4.b.c 24
12.b even 2 1 36.4.h.b ✓ 24
36.f odd 6 1 36.4.h.b ✓ 24
36.f odd 6 1 324.4.b.c 24
36.h even 6 1 inner 108.4.h.b 24
36.h even 6 1 324.4.b.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.h.b ✓ 24 3.b odd 2 1
36.4.h.b ✓ 24 9.c even 3 1
36.4.h.b ✓ 24 12.b even 2 1
36.4.h.b ✓ 24 36.f odd 6 1
108.4.h.b 24 1.a even 1 1 trivial
108.4.h.b 24 4.b odd 2 1 inner
108.4.h.b 24 9.d odd 6 1 inner
108.4.h.b 24 36.h even 6 1 inner
324.4.b.c 24 9.c even 3 1
324.4.b.c 24 9.d odd 6 1
324.4.b.c 24 36.f odd 6 1
324.4.b.c 24 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 36 T_{5}^{11} + 210 T_{5}^{10} + 7992 T_{5}^{9} - 59073 T_{5}^{8} - 2269332 T_{5}^{7} + \cdots + 7678666384 \) acting on \(S_{4}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display