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Properties

Label 216.4.l.b
Level $216$
Weight $4$
Character orbit 216.l
Analytic conductor $12.744$
Analytic rank $0$
Dimension $64$
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] = [216,4,Mod(35,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("216.35");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.l (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: \(12.7444125612\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 72)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 3 q^{2} - 17 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 3 q^{2} - 17 q^{4} + 12 q^{10} - 48 q^{11} - 72 q^{14} + 127 q^{16} - 220 q^{19} + 234 q^{20} - 217 q^{22} - 902 q^{25} - 132 q^{28} + 693 q^{32} + 509 q^{34} + 1977 q^{38} - 36 q^{40} - 1620 q^{41} - 292 q^{43} + 48 q^{46} + 1762 q^{49} + 1227 q^{50} + 330 q^{52} - 942 q^{56} - 282 q^{58} - 5592 q^{59} + 1090 q^{64} + 6 q^{65} + 68 q^{67} + 2025 q^{68} + 600 q^{70} - 868 q^{73} + 420 q^{74} - 1471 q^{76} + 362 q^{82} - 3654 q^{83} + 4119 q^{86} + 3155 q^{88} - 1380 q^{91} + 744 q^{92} - 138 q^{94} - 1912 q^{97}+O(q^{100}) \) 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 −2.82824 + 0.0321058i 0 7.99794 − 0.181606i 6.78045 + 11.7441i 0 −14.1027 − 8.14219i −22.6143 + 0.770407i 0 −19.5538 − 32.9975i
35.2 −2.82386 − 0.160583i 0 7.94843 + 0.906929i −5.34568 − 9.25900i 0 15.5169 + 8.95867i −22.2996 − 3.83743i 0 13.6087 + 27.0046i
35.3 −2.78409 + 0.498867i 0 7.50226 − 2.77778i 1.46312 + 2.53420i 0 −4.48829 − 2.59132i −19.5012 + 11.4762i 0 −5.33769 − 6.32553i
35.4 −2.53000 − 1.26456i 0 4.80175 + 6.39868i −8.03411 − 13.9155i 0 −29.5192 − 17.0429i −4.05686 − 22.2608i 0 2.72923 + 45.3658i
35.5 −2.42735 + 1.45189i 0 3.78402 − 7.04849i −4.95891 − 8.58909i 0 11.4514 + 6.61147i 1.04853 + 22.6031i 0 24.5074 + 13.6489i
35.6 −2.36014 − 1.55876i 0 3.14055 + 7.35778i 8.03411 + 13.9155i 0 29.5192 + 17.0429i 4.05686 − 22.2608i 0 2.72923 − 45.3658i
35.7 −2.31471 + 1.62546i 0 2.71576 − 7.52494i 10.8653 + 18.8192i 0 10.5583 + 6.09581i 5.94529 + 21.8324i 0 −55.7398 − 25.8999i
35.8 −1.89798 + 2.09706i 0 −0.795355 − 7.96036i −8.51655 − 14.7511i 0 9.57807 + 5.52990i 18.2030 + 13.4407i 0 47.0983 + 10.1375i
35.9 −1.64607 + 2.30010i 0 −2.58093 − 7.57224i −2.83055 − 4.90265i 0 −25.8587 − 14.9296i 21.6653 + 6.52801i 0 15.9359 + 1.55955i
35.10 −1.55100 − 2.36525i 0 −3.18879 + 7.33700i 5.34568 + 9.25900i 0 −15.5169 − 8.95867i 22.2996 − 3.83743i 0 13.6087 − 27.0046i
35.11 −1.38632 − 2.46538i 0 −4.15624 + 6.83561i −6.78045 − 11.7441i 0 14.1027 + 8.14219i 22.6143 + 0.770407i 0 −19.5538 + 32.9975i
35.12 −1.35255 + 2.48407i 0 −4.34119 − 6.71968i 3.60771 + 6.24874i 0 −6.44991 − 3.72386i 22.5638 − 1.69509i 0 −20.4019 + 0.510040i
35.13 −0.960011 − 2.66052i 0 −6.15676 + 5.10826i −1.46312 − 2.53420i 0 4.48829 + 2.59132i 19.5012 + 11.4762i 0 −5.33769 + 6.32553i
35.14 −0.332650 + 2.80880i 0 −7.77869 − 1.86869i 4.74971 + 8.22674i 0 16.7522 + 9.67189i 7.83636 − 21.2271i 0 −24.6872 + 10.6043i
35.15 0.0437026 − 2.82809i 0 −7.99618 − 0.247190i 4.95891 + 8.58909i 0 −11.4514 − 6.61147i −1.04853 + 22.6031i 0 24.5074 − 13.6489i
35.16 0.229927 + 2.81907i 0 −7.89427 + 1.29636i 2.33412 + 4.04281i 0 −28.2737 − 16.3238i −5.46963 − 21.9564i 0 −10.8603 + 7.50958i
35.17 0.232248 + 2.81888i 0 −7.89212 + 1.30935i −1.01248 − 1.75366i 0 16.3356 + 9.43138i −5.52384 − 21.9428i 0 4.70820 − 3.26133i
35.18 0.250335 − 2.81733i 0 −7.87466 − 1.41055i −10.8653 − 18.8192i 0 −10.5583 − 6.09581i −5.94529 + 21.8324i 0 −55.7398 + 25.8999i
35.19 0.616605 + 2.76040i 0 −7.23960 + 3.40415i −8.75283 − 15.1603i 0 14.7005 + 8.48731i −13.8608 − 17.8852i 0 36.4516 − 33.5092i
35.20 0.867121 − 2.69223i 0 −6.49620 − 4.66898i 8.51655 + 14.7511i 0 −9.57807 − 5.52990i −18.2030 + 13.4407i 0 47.0983 − 10.1375i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.4.l.b 64
3.b odd 2 1 72.4.l.b ✓ 64
4.b odd 2 1 864.4.p.b 64
8.b even 2 1 864.4.p.b 64
8.d odd 2 1 inner 216.4.l.b 64
9.c even 3 1 72.4.l.b ✓ 64
9.d odd 6 1 inner 216.4.l.b 64
12.b even 2 1 288.4.p.b 64
24.f even 2 1 72.4.l.b ✓ 64
24.h odd 2 1 288.4.p.b 64
36.f odd 6 1 288.4.p.b 64
36.h even 6 1 864.4.p.b 64
72.j odd 6 1 864.4.p.b 64
72.l even 6 1 inner 216.4.l.b 64
72.n even 6 1 288.4.p.b 64
72.p odd 6 1 72.4.l.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.l.b ✓ 64 3.b odd 2 1
72.4.l.b ✓ 64 9.c even 3 1
72.4.l.b ✓ 64 24.f even 2 1
72.4.l.b ✓ 64 72.p odd 6 1
216.4.l.b 64 1.a even 1 1 trivial
216.4.l.b 64 8.d odd 2 1 inner
216.4.l.b 64 9.d odd 6 1 inner
216.4.l.b 64 72.l even 6 1 inner
288.4.p.b 64 12.b even 2 1
288.4.p.b 64 24.h odd 2 1
288.4.p.b 64 36.f odd 6 1
288.4.p.b 64 72.n even 6 1
864.4.p.b 64 4.b odd 2 1
864.4.p.b 64 8.b even 2 1
864.4.p.b 64 36.h even 6 1
864.4.p.b 64 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} + 2451 T_{5}^{62} + 3328326 T_{5}^{60} + 3108844719 T_{5}^{58} + 2205690843492 T_{5}^{56} + \cdots + 49\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display