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Properties

Label 81.5.f.a
Level $81$
Weight $5$
Character orbit 81.f
Analytic conductor $8.373$
Analytic rank $0$
Dimension $66$
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] = [81,5,Mod(8,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.8");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 81.f (of order \(18\), degree \(6\), 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: \(8.37296700979\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} + 492 q^{11} - 6 q^{13} - 1137 q^{14} - 54 q^{16} + 9 q^{17} - 3 q^{19} + 2487 q^{20} + 1002 q^{22} + 2724 q^{23} + 435 q^{25} - 12 q^{28} - 5016 q^{29} - 1671 q^{31} - 10224 q^{32} - 342 q^{34} + 2682 q^{35} - 3 q^{37} + 6564 q^{38} - 1113 q^{40} - 5754 q^{41} - 1266 q^{43} + 4545 q^{44} - 3 q^{46} + 9159 q^{47} + 5898 q^{49} + 34977 q^{50} + 2871 q^{52} - 12 q^{55} - 39243 q^{56} - 12291 q^{58} - 24762 q^{59} - 8358 q^{61} - 38304 q^{62} + 6141 q^{64} - 23727 q^{65} + 15996 q^{67} + 43533 q^{68} - 7251 q^{70} + 19773 q^{71} + 6108 q^{73} + 74847 q^{74} - 28614 q^{76} + 33909 q^{77} - 5658 q^{79} - 12 q^{82} - 18813 q^{83} + 24219 q^{85} - 96474 q^{86} + 36042 q^{88} - 110232 q^{89} - 6042 q^{91} - 73545 q^{92} + 2631 q^{94} + 65163 q^{95} - 3696 q^{97} + 259938 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −7.34334 − 1.29483i 0 37.2130 + 13.5444i 22.9421 + 27.3414i 0 14.0427 − 5.11111i −152.408 − 87.9929i 0 −133.070 − 230.483i
8.2 −5.98494 − 1.05531i 0 19.6707 + 7.15955i −10.5942 − 12.6257i 0 7.09675 − 2.58301i −25.9633 − 14.9899i 0 50.0816 + 86.7438i
8.3 −4.49504 − 0.792597i 0 4.54211 + 1.65319i 3.40574 + 4.05880i 0 −28.7185 + 10.4527i 44.1393 + 25.4838i 0 −12.0919 − 20.9439i
8.4 −2.89555 − 0.510564i 0 −6.91152 − 2.51559i −31.8237 − 37.9260i 0 28.2667 − 10.2882i 59.4692 + 34.3346i 0 72.7837 + 126.065i
8.5 −1.18141 − 0.208314i 0 −13.6827 − 4.98011i 12.1061 + 14.4275i 0 0.372391 − 0.135539i 31.7501 + 18.3309i 0 −11.2968 − 19.5667i
8.6 0.185922 + 0.0327832i 0 −15.0016 − 5.46013i 10.5929 + 12.6241i 0 26.3432 − 9.58813i −5.22609 − 3.01729i 0 1.55560 + 2.69438i
8.7 2.38785 + 0.421042i 0 −9.51055 − 3.46156i 12.2061 + 14.5467i 0 −77.2633 + 28.1215i −54.8497 − 31.6675i 0 23.0216 + 39.8746i
8.8 3.58910 + 0.632855i 0 −2.55397 − 0.929568i −12.8585 − 15.3241i 0 75.5885 − 27.5120i −59.0773 − 34.1083i 0 −36.4523 − 63.1373i
8.9 4.35935 + 0.768672i 0 3.37802 + 1.22950i −27.2417 − 32.4654i 0 −49.1081 + 17.8739i −47.5559 − 27.4564i 0 −93.8009 − 162.468i
8.10 6.19782 + 1.09284i 0 22.1836 + 8.07418i 7.06026 + 8.41408i 0 62.3269 − 22.6852i 41.4621 + 23.9381i 0 34.5629 + 59.8648i
8.11 7.11993 + 1.25544i 0 34.0822 + 12.4049i 16.3502 + 19.4854i 0 −60.7132 + 22.0978i 126.911 + 73.2722i 0 91.9496 + 159.261i
17.1 −2.52261 + 6.93081i 0 −29.4158 − 24.6828i −21.3946 − 3.77244i 0 −45.7826 + 38.4162i 143.077 − 82.6054i 0 80.1161 − 138.765i
17.2 −2.28985 + 6.29131i 0 −22.0805 − 18.5277i 25.7110 + 4.53354i 0 68.4999 − 57.4782i 74.3553 − 42.9290i 0 −87.3962 + 151.375i
17.3 −1.62498 + 4.46460i 0 −5.03539 − 4.22519i 1.85566 + 0.327202i 0 −15.2767 + 12.8186i −38.7874 + 22.3939i 0 −4.47623 + 7.75306i
17.4 −0.940198 + 2.58317i 0 6.46790 + 5.42721i −20.2618 − 3.57269i 0 −3.42243 + 2.87176i −58.1912 + 33.5967i 0 28.2789 − 48.9806i
17.5 −0.498289 + 1.36904i 0 10.6307 + 8.92025i 15.7451 + 2.77629i 0 22.4150 − 18.8084i −37.6967 + 21.7642i 0 −11.6465 + 20.1723i
17.6 0.254601 − 0.699511i 0 11.8322 + 9.92841i 32.8927 + 5.79987i 0 −64.8213 + 54.3915i 20.2723 − 11.7042i 0 12.4316 − 21.5322i
17.7 0.410788 − 1.12863i 0 11.1517 + 9.35735i −35.6423 − 6.28469i 0 50.7952 − 42.6222i 31.7844 − 18.3507i 0 −21.7345 + 37.6453i
17.8 1.40615 − 3.86336i 0 −0.691617 − 0.580335i 11.8424 + 2.08814i 0 13.5416 − 11.3628i 53.7534 − 31.0345i 0 24.7194 − 42.8153i
17.9 1.54114 − 4.23424i 0 −3.29700 − 2.76651i −31.7277 − 5.59444i 0 −45.5870 + 38.2520i 45.6416 − 26.3512i 0 −72.5850 + 125.721i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.11
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 81.5.f.a 66
3.b odd 2 1 27.5.f.a ✓ 66
27.e even 9 1 27.5.f.a ✓ 66
27.f odd 18 1 inner 81.5.f.a 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.5.f.a ✓ 66 3.b odd 2 1
27.5.f.a ✓ 66 27.e even 9 1
81.5.f.a 66 1.a even 1 1 trivial
81.5.f.a 66 27.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(81, [\chi])\).