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Properties

Label 81.3.f.a
Level $81$
Weight $3$
Character orbit 81.f
Analytic conductor $2.207$
Analytic rank $0$
Dimension $30$
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] = [81,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.8");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
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: \(2.20709014132\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 6 q^{4} + 15 q^{5} - 6 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 6 q^{4} + 15 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} + 6 q^{11} - 6 q^{13} + 15 q^{14} - 18 q^{16} + 9 q^{17} - 3 q^{19} - 213 q^{20} - 42 q^{22} - 120 q^{23} - 15 q^{25} - 12 q^{28} + 168 q^{29} + 39 q^{31} + 360 q^{32} + 54 q^{34} + 252 q^{35} - 3 q^{37} + 84 q^{38} - 33 q^{40} - 228 q^{41} - 96 q^{43} - 639 q^{44} - 3 q^{46} - 399 q^{47} - 78 q^{49} - 303 q^{50} - 9 q^{52} - 12 q^{55} + 393 q^{56} + 129 q^{58} + 474 q^{59} + 138 q^{61} + 900 q^{62} - 51 q^{64} + 411 q^{65} + 354 q^{67} - 99 q^{68} + 489 q^{70} - 315 q^{71} - 66 q^{73} - 321 q^{74} + 258 q^{76} - 201 q^{77} + 30 q^{79} - 12 q^{82} + 33 q^{83} - 261 q^{85} + 258 q^{86} - 642 q^{88} - 72 q^{89} + 96 q^{91} + 3 q^{92} - 861 q^{94} - 681 q^{95} - 582 q^{97} - 882 q^{98}+O(q^{100}) \) 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 −2.58003 − 0.454929i 0 2.69082 + 0.979377i −0.519123 − 0.618667i 0 −5.56035 + 2.02380i 2.57852 + 1.48871i 0 1.05790 + 1.83234i
8.2 −1.14332 − 0.201599i 0 −2.49222 − 0.907094i 3.46013 + 4.12362i 0 9.89907 − 3.60297i 6.68824 + 3.86146i 0 −3.12473 − 5.41219i
8.3 −0.115908 − 0.0204377i 0 −3.74575 − 1.36334i −3.98394 − 4.74788i 0 −7.49258 + 2.72708i 0.814011 + 0.469969i 0 0.364735 + 0.631740i
8.4 2.31604 + 0.408381i 0 1.43851 + 0.523575i 3.71692 + 4.42965i 0 4.57693 − 1.66587i −5.02894 − 2.90346i 0 6.79956 + 11.7772i
8.5 3.46291 + 0.610605i 0 7.86014 + 2.86086i −3.16671 − 3.77394i 0 −3.18911 + 1.16074i 13.2912 + 7.67367i 0 −8.66165 − 15.0024i
17.1 −0.898786 + 2.46939i 0 −2.22591 − 1.86776i −6.15614 − 1.08549i 0 −6.21846 + 5.21791i −2.49037 + 1.43782i 0 8.21356 − 14.2263i
17.2 −0.833352 + 2.28962i 0 −1.48369 − 1.24496i 8.90104 + 1.56949i 0 −1.32607 + 1.11270i −4.35357 + 2.51353i 0 −11.0112 + 19.0720i
17.3 0.155957 − 0.428490i 0 2.90490 + 2.43750i 2.41673 + 0.426135i 0 −3.31426 + 2.78099i 3.07708 − 1.77655i 0 0.559502 − 0.969085i
17.4 0.524483 − 1.44101i 0 1.26276 + 1.05958i 0.0496479 + 0.00875427i 0 7.55168 − 6.33661i 7.50132 − 4.33089i 0 0.0386545 − 0.0669515i
17.5 1.28565 − 3.53230i 0 −7.76007 − 6.51147i 0.526552 + 0.0928453i 0 2.13346 − 1.79019i −19.9557 + 11.5214i 0 1.00492 − 1.74057i
35.1 −2.13670 + 2.54642i 0 −1.22417 − 6.94260i 2.35247 − 6.46335i 0 1.10811 − 6.28443i 8.77937 + 5.06877i 0 11.4319 + 19.8006i
35.2 −0.837612 + 0.998227i 0 0.399729 + 2.26698i −0.149473 + 0.410673i 0 −1.05651 + 5.99176i −7.11182 − 4.10601i 0 −0.284745 − 0.493192i
35.3 0.374063 − 0.445791i 0 0.635786 + 3.60572i −2.62195 + 7.20376i 0 0.231638 − 1.31369i 3.86112 + 2.22922i 0 2.23059 + 3.86350i
35.4 1.24712 − 1.48626i 0 0.0409354 + 0.232156i 1.47839 − 4.06185i 0 1.54363 − 8.75434i 7.11704 + 4.10903i 0 −4.19322 − 7.26288i
35.5 2.17948 − 2.59740i 0 −1.30178 − 7.38274i 1.19547 − 3.28452i 0 −1.88718 + 10.7027i −10.2675 − 5.92795i 0 −5.92572 − 10.2637i
44.1 −2.13670 − 2.54642i 0 −1.22417 + 6.94260i 2.35247 + 6.46335i 0 1.10811 + 6.28443i 8.77937 − 5.06877i 0 11.4319 − 19.8006i
44.2 −0.837612 − 0.998227i 0 0.399729 − 2.26698i −0.149473 − 0.410673i 0 −1.05651 − 5.99176i −7.11182 + 4.10601i 0 −0.284745 + 0.493192i
44.3 0.374063 + 0.445791i 0 0.635786 − 3.60572i −2.62195 − 7.20376i 0 0.231638 + 1.31369i 3.86112 − 2.22922i 0 2.23059 − 3.86350i
44.4 1.24712 + 1.48626i 0 0.0409354 − 0.232156i 1.47839 + 4.06185i 0 1.54363 + 8.75434i 7.11704 − 4.10903i 0 −4.19322 + 7.26288i
44.5 2.17948 + 2.59740i 0 −1.30178 + 7.38274i 1.19547 + 3.28452i 0 −1.88718 − 10.7027i −10.2675 + 5.92795i 0 −5.92572 + 10.2637i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.5
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 81.3.f.a 30
3.b odd 2 1 27.3.f.a ✓ 30
9.c even 3 1 243.3.f.a 30
9.c even 3 1 243.3.f.b 30
9.d odd 6 1 243.3.f.c 30
9.d odd 6 1 243.3.f.d 30
12.b even 2 1 432.3.bc.a 30
27.e even 9 1 27.3.f.a ✓ 30
27.e even 9 1 243.3.f.c 30
27.e even 9 1 243.3.f.d 30
27.e even 9 1 729.3.b.a 30
27.f odd 18 1 inner 81.3.f.a 30
27.f odd 18 1 243.3.f.a 30
27.f odd 18 1 243.3.f.b 30
27.f odd 18 1 729.3.b.a 30
108.j odd 18 1 432.3.bc.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.f.a ✓ 30 3.b odd 2 1
27.3.f.a ✓ 30 27.e even 9 1
81.3.f.a 30 1.a even 1 1 trivial
81.3.f.a 30 27.f odd 18 1 inner
243.3.f.a 30 9.c even 3 1
243.3.f.a 30 27.f odd 18 1
243.3.f.b 30 9.c even 3 1
243.3.f.b 30 27.f odd 18 1
243.3.f.c 30 9.d odd 6 1
243.3.f.c 30 27.e even 9 1
243.3.f.d 30 9.d odd 6 1
243.3.f.d 30 27.e even 9 1
432.3.bc.a 30 12.b even 2 1
432.3.bc.a 30 108.j odd 18 1
729.3.b.a 30 27.e even 9 1
729.3.b.a 30 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(81, [\chi])\).