Properties

Label 216.4.n.a
Level $216$
Weight $4$
Character orbit 216.n
Analytic conductor $12.744$
Analytic rank $0$
Dimension $68$
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(37,216)]
 
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([0, 3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.n (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: \(68\)
Relative dimension: \(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + q^{2} - q^{4} - 2 q^{7} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + q^{2} - q^{4} - 2 q^{7} + 10 q^{8} - 20 q^{10} + 10 q^{14} - q^{16} + 8 q^{17} + 52 q^{20} - 17 q^{22} - 274 q^{23} + 648 q^{25} - 368 q^{26} + 124 q^{28} - 2 q^{31} - 259 q^{32} + 189 q^{34} - 319 q^{38} + 214 q^{40} + 22 q^{41} - 282 q^{44} - 24 q^{46} + 942 q^{47} - 1080 q^{49} - 53 q^{50} - 588 q^{52} - 508 q^{55} + 502 q^{56} + 280 q^{58} - 1744 q^{62} + 410 q^{64} + 502 q^{65} - 1149 q^{68} - 586 q^{70} + 3984 q^{71} - 8 q^{73} - 1778 q^{74} + 621 q^{76} - 2 q^{79} - 4704 q^{80} + 714 q^{82} + 2923 q^{86} - 533 q^{88} + 856 q^{89} - 3342 q^{92} + 1518 q^{94} + 2792 q^{95} - 2 q^{97} + 6414 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −2.82826 0.0306697i 0 7.99812 + 0.173484i 4.46589 + 2.57838i 0 −4.47414 7.74943i −22.6154 0.735958i 0 −12.5516 7.42931i
37.2 −2.81886 0.232479i 0 7.89191 + 1.31065i −2.06876 1.19440i 0 17.9335 + 31.0618i −21.9415 5.52925i 0 5.55387 + 3.84779i
37.3 −2.80631 + 0.353019i 0 7.75076 1.98136i −13.3655 7.71659i 0 −14.8241 25.6761i −21.0516 + 8.29647i 0 40.2319 + 16.9369i
37.4 −2.67089 + 0.930772i 0 6.26733 4.97198i 14.6790 + 8.47493i 0 9.54171 + 16.5267i −12.1116 + 19.1131i 0 −47.0943 8.97282i
37.5 −2.48873 1.34396i 0 4.38752 + 6.68952i −6.59200 3.80589i 0 −7.67810 13.2989i −1.92888 22.5451i 0 11.2907 + 18.3312i
37.6 −2.34837 + 1.57644i 0 3.02970 7.40412i −6.66588 3.84855i 0 4.44653 + 7.70162i 4.55727 + 22.1637i 0 21.7209 1.47052i
37.7 −2.30783 1.63521i 0 2.65219 + 7.54757i −3.38868 1.95645i 0 1.48218 + 2.56721i 6.22103 21.7554i 0 4.62129 + 10.0564i
37.8 −2.17295 + 1.81060i 0 1.44346 7.86870i 7.42736 + 4.28819i 0 −10.3525 17.9310i 11.1105 + 19.7119i 0 −23.9035 + 4.12993i
37.9 −2.16881 1.81556i 0 1.40750 + 7.87521i 17.5797 + 10.1497i 0 −10.6896 18.5150i 11.2453 19.6353i 0 −19.6999 53.9298i
37.10 −1.77410 + 2.20286i 0 −1.70517 7.81616i −13.2040 7.62335i 0 6.18711 + 10.7164i 20.2430 + 10.1104i 0 40.2184 15.5620i
37.11 −1.59019 2.33908i 0 −2.94259 + 7.43916i 11.0897 + 6.40265i 0 11.0213 + 19.0894i 22.0801 4.94672i 0 −2.65844 36.1212i
37.12 −1.23061 2.54668i 0 −4.97121 + 6.26794i −11.0897 6.40265i 0 11.0213 + 19.0894i 22.0801 + 4.94672i 0 −2.65844 + 36.1212i
37.13 −1.08796 + 2.61081i 0 −5.63270 5.68091i 10.5646 + 6.09948i 0 −9.53003 16.5065i 20.9599 8.52535i 0 −27.4185 + 20.9463i
37.14 −0.833385 + 2.70286i 0 −6.61094 4.50505i 2.76156 + 1.59439i 0 1.73091 + 2.99802i 17.6860 14.1140i 0 −6.61085 + 6.13538i
37.15 −0.487912 2.78603i 0 −7.52388 + 2.71867i −17.5797 10.1497i 0 −10.6896 18.5150i 11.2453 + 19.6353i 0 −19.6999 + 53.9298i
37.16 −0.262215 2.81625i 0 −7.86249 + 1.47692i 3.38868 + 1.95645i 0 1.48218 + 2.56721i 6.22103 + 21.7554i 0 4.62129 10.0564i
37.17 −0.181886 + 2.82257i 0 −7.93383 1.02677i 13.2324 + 7.63974i 0 11.2586 + 19.5004i 4.34120 22.2071i 0 −23.9705 + 35.9599i
37.18 −0.176838 + 2.82289i 0 −7.93746 0.998392i −17.3643 10.0253i 0 1.69820 + 2.94137i 4.22200 22.2300i 0 31.3710 47.2447i
37.19 0.0804569 2.82728i 0 −7.98705 0.454949i 6.59200 + 3.80589i 0 −7.67810 13.2989i −1.92888 + 22.5451i 0 11.2907 18.3312i
37.20 0.980526 + 2.65303i 0 −6.07714 + 5.20273i −0.772898 0.446233i 0 −14.6308 25.3412i −19.7618 11.0214i 0 0.426023 2.48806i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n 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.n.a 68
3.b odd 2 1 72.4.n.a 68
4.b odd 2 1 864.4.r.a 68
8.b even 2 1 inner 216.4.n.a 68
8.d odd 2 1 864.4.r.a 68
9.c even 3 1 inner 216.4.n.a 68
9.d odd 6 1 72.4.n.a 68
12.b even 2 1 288.4.r.a 68
24.f even 2 1 288.4.r.a 68
24.h odd 2 1 72.4.n.a 68
36.f odd 6 1 864.4.r.a 68
36.h even 6 1 288.4.r.a 68
72.j odd 6 1 72.4.n.a 68
72.l even 6 1 288.4.r.a 68
72.n even 6 1 inner 216.4.n.a 68
72.p odd 6 1 864.4.r.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.n.a 68 3.b odd 2 1
72.4.n.a 68 9.d odd 6 1
72.4.n.a 68 24.h odd 2 1
72.4.n.a 68 72.j odd 6 1
216.4.n.a 68 1.a even 1 1 trivial
216.4.n.a 68 8.b even 2 1 inner
216.4.n.a 68 9.c even 3 1 inner
216.4.n.a 68 72.n even 6 1 inner
288.4.r.a 68 12.b even 2 1
288.4.r.a 68 24.f even 2 1
288.4.r.a 68 36.h even 6 1
288.4.r.a 68 72.l even 6 1
864.4.r.a 68 4.b odd 2 1
864.4.r.a 68 8.d odd 2 1
864.4.r.a 68 36.f odd 6 1
864.4.r.a 68 72.p odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(216, [\chi])\).