↑

Properties

Label 972.2.l.d
Level $972$
Weight $2$
Character orbit 972.l
Analytic conductor $7.761$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [972,2,Mod(107,972)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(972, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("972.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 972 = 2^{2} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 972.l (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: \(7.76145907647\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 9 q^{8} - 3 q^{10} + 6 q^{13} + 33 q^{14} + 3 q^{16} - 18 q^{17} - 18 q^{20} + 3 q^{22} + 6 q^{25} - 12 q^{28} + 30 q^{29} + 33 q^{32} + 15 q^{34} - 6 q^{37} - 63 q^{38} + 33 q^{40} - 24 q^{41} + 63 q^{44} - 3 q^{46} + 6 q^{49} - 21 q^{50} + 57 q^{52} - 18 q^{56} + 57 q^{58} + 6 q^{61} + 90 q^{62} - 3 q^{64} + 30 q^{65} - 102 q^{68} + 51 q^{70} - 6 q^{73} + 75 q^{74} + 27 q^{76} - 42 q^{77} - 12 q^{82} - 24 q^{85} - 111 q^{86} + 27 q^{88} + 147 q^{92} + 30 q^{94} - 12 q^{97} - 180 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.41292 + 0.0603944i 0 1.99271 − 0.170665i −0.0476648 + 0.130958i 0 4.57823 + 0.807266i −2.80523 + 0.361485i 0 0.0594376 − 0.187912i
107.2 −1.38367 + 0.292332i 0 1.82908 − 0.808982i −1.15402 + 3.17063i 0 −2.64500 − 0.466385i −2.29436 + 1.65406i 0 0.669899 − 4.72446i
107.3 −1.27919 − 0.603048i 0 1.27267 + 1.54283i −0.103444 + 0.284210i 0 0.478118 + 0.0843052i −0.697584 − 2.74105i 0 0.303717 − 0.301177i
107.4 −1.14898 + 0.824528i 0 0.640306 − 1.89473i 0.840877 − 2.31029i 0 −3.82489 − 0.674430i 0.826562 + 2.70496i 0 0.938750 + 3.34780i
107.5 −1.09740 − 0.892026i 0 0.408579 + 1.95782i 1.17862 − 3.23823i 0 −2.88522 − 0.508742i 1.29805 − 2.51298i 0 −4.18200 + 2.50228i
107.6 −0.612484 + 1.27470i 0 −1.24973 − 1.56147i 0.840877 − 2.31029i 0 3.82489 + 0.674430i 2.75584 − 0.636655i 0 2.42991 + 2.48688i
107.7 −0.374630 − 1.36369i 0 −1.71931 + 1.02176i −0.184791 + 0.507709i 0 1.81997 + 0.320909i 2.03746 + 1.96182i 0 0.761586 + 0.0617949i
107.8 −0.0476191 + 1.41341i 0 −1.99546 − 0.134611i −1.15402 + 3.17063i 0 2.64500 + 0.466385i 0.285283 − 2.81400i 0 −4.42645 − 1.78208i
107.9 0.185875 + 1.40195i 0 −1.93090 + 0.521172i −0.0476648 + 0.130958i 0 −4.57823 − 0.807266i −1.08956 − 2.61015i 0 −0.192456 − 0.0424817i
107.10 0.419187 − 1.35066i 0 −1.64856 − 1.13236i −0.776998 + 2.13478i 0 −1.29035 − 0.227523i −2.22049 + 1.75198i 0 2.55766 + 1.94434i
107.11 0.613200 − 1.27436i 0 −1.24797 − 1.56287i 0.921065 − 2.53060i 0 −0.686687 − 0.121081i −2.75691 + 0.632009i 0 −2.66010 − 2.72553i
107.12 0.816016 + 1.15504i 0 −0.668236 + 1.88506i −0.103444 + 0.284210i 0 −0.478118 − 0.0843052i −2.72261 + 0.766402i 0 −0.412685 + 0.112438i
107.13 1.06904 + 0.925831i 0 0.285675 + 1.97949i 1.17862 − 3.23823i 0 2.88522 + 0.508742i −1.52728 + 2.38064i 0 4.25804 − 2.37058i
107.14 1.14852 − 0.825174i 0 0.638176 − 1.89545i 0.921065 − 2.53060i 0 0.686687 + 0.121081i −0.831120 − 2.70356i 0 −1.03033 − 3.66648i
107.15 1.25735 − 0.647359i 0 1.16185 − 1.62791i −0.776998 + 2.13478i 0 1.29035 + 0.227523i 0.407013 − 2.79899i 0 0.405013 + 3.18717i
107.16 1.40803 + 0.132136i 0 1.96508 + 0.372101i −0.184791 + 0.507709i 0 −1.81997 − 0.320909i 2.71772 + 0.783586i 0 −0.327277 + 0.690450i
215.1 −1.40391 − 0.170372i 0 1.94195 + 0.478375i −0.371030 + 0.442176i 0 −0.592856 + 1.62886i −2.64482 − 1.00245i 0 0.596228 − 0.557564i
215.2 −1.26251 − 0.637243i 0 1.18784 + 1.60905i 0.883505 − 1.05292i 0 0.546743 − 1.50217i −0.474302 − 2.78838i 0 −1.78640 + 0.766310i
215.3 −1.26098 + 0.640264i 0 1.18012 − 1.61472i −0.371030 + 0.442176i 0 0.592856 − 1.62886i −0.454264 + 2.79171i 0 0.184750 − 0.795131i
215.4 −0.968417 + 1.03062i 0 −0.124336 − 1.99613i 0.883505 − 1.05292i 0 −0.546743 + 1.50217i 2.17765 + 1.80495i 0 0.229554 + 1.93022i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 972.2.l.d 96
3.b odd 2 1 972.2.l.a 96
4.b odd 2 1 inner 972.2.l.d 96
9.c even 3 1 108.2.l.a ✓ 96
9.c even 3 1 972.2.l.c 96
9.d odd 6 1 324.2.l.a 96
9.d odd 6 1 972.2.l.b 96
12.b even 2 1 972.2.l.a 96
27.e even 9 1 324.2.l.a 96
27.e even 9 1 972.2.l.a 96
27.e even 9 1 972.2.l.b 96
27.f odd 18 1 108.2.l.a ✓ 96
27.f odd 18 1 972.2.l.c 96
27.f odd 18 1 inner 972.2.l.d 96
36.f odd 6 1 108.2.l.a ✓ 96
36.f odd 6 1 972.2.l.c 96
36.h even 6 1 324.2.l.a 96
36.h even 6 1 972.2.l.b 96
108.j odd 18 1 324.2.l.a 96
108.j odd 18 1 972.2.l.a 96
108.j odd 18 1 972.2.l.b 96
108.l even 18 1 108.2.l.a ✓ 96
108.l even 18 1 972.2.l.c 96
108.l even 18 1 inner 972.2.l.d 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.l.a ✓ 96 9.c even 3 1
108.2.l.a ✓ 96 27.f odd 18 1
108.2.l.a ✓ 96 36.f odd 6 1
108.2.l.a ✓ 96 108.l even 18 1
324.2.l.a 96 9.d odd 6 1
324.2.l.a 96 27.e even 9 1
324.2.l.a 96 36.h even 6 1
324.2.l.a 96 108.j odd 18 1
972.2.l.a 96 3.b odd 2 1
972.2.l.a 96 12.b even 2 1
972.2.l.a 96 27.e even 9 1
972.2.l.a 96 108.j odd 18 1
972.2.l.b 96 9.d odd 6 1
972.2.l.b 96 27.e even 9 1
972.2.l.b 96 36.h even 6 1
972.2.l.b 96 108.j odd 18 1
972.2.l.c 96 9.c even 3 1
972.2.l.c 96 27.f odd 18 1
972.2.l.c 96 36.f odd 6 1
972.2.l.c 96 108.l even 18 1
972.2.l.d 96 1.a even 1 1 trivial
972.2.l.d 96 4.b odd 2 1 inner
972.2.l.d 96 27.f odd 18 1 inner
972.2.l.d 96 108.l even 18 1 inner