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Properties

Label 972.2.l.a
Level $972$
Weight $2$
Character orbit 972.l
Analytic conductor $7.761$
Analytic rank $0$
Dimension $96$
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] = [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 −1.40803 − 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
107.2 −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
107.3 −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
107.4 −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
107.5 −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
107.6 −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
107.7 −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
107.8 −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
107.9 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
107.10 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
107.11 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
107.12 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
107.13 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
107.14 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
107.15 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
107.16 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
215.1 −1.40942 + 0.116308i 0 1.97294 − 0.327854i 1.33487 − 1.59083i 0 −1.33353 + 3.66385i −2.74258 + 0.691554i 0 −1.69636 + 2.39741i
215.2 −1.36870 − 0.355894i 0 1.74668 + 0.974223i −2.43805 + 2.90555i 0 0.896181 − 2.46224i −2.04396 − 1.95505i 0 4.37103 − 3.10915i
215.3 −1.36420 + 0.372757i 0 1.72210 − 1.01703i 1.33487 − 1.59083i 0 1.33353 − 3.66385i −1.97019 + 2.02937i 0 −1.22804 + 2.66780i
215.4 −1.16443 + 0.802554i 0 0.711816 − 1.86904i −2.43805 + 2.90555i 0 −0.896181 + 2.46224i 0.671144 + 2.74765i 0 0.507086 − 5.33999i
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.a 96
3.b odd 2 1 972.2.l.d 96
4.b odd 2 1 inner 972.2.l.a 96
9.c even 3 1 324.2.l.a 96
9.c even 3 1 972.2.l.b 96
9.d odd 6 1 108.2.l.a ✓ 96
9.d odd 6 1 972.2.l.c 96
12.b even 2 1 972.2.l.d 96
27.e even 9 1 108.2.l.a ✓ 96
27.e even 9 1 972.2.l.c 96
27.e even 9 1 972.2.l.d 96
27.f odd 18 1 324.2.l.a 96
27.f odd 18 1 inner 972.2.l.a 96
27.f odd 18 1 972.2.l.b 96
36.f odd 6 1 324.2.l.a 96
36.f odd 6 1 972.2.l.b 96
36.h even 6 1 108.2.l.a ✓ 96
36.h even 6 1 972.2.l.c 96
108.j odd 18 1 108.2.l.a ✓ 96
108.j odd 18 1 972.2.l.c 96
108.j odd 18 1 972.2.l.d 96
108.l even 18 1 324.2.l.a 96
108.l even 18 1 inner 972.2.l.a 96
108.l even 18 1 972.2.l.b 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.l.a ✓ 96 9.d odd 6 1
108.2.l.a ✓ 96 27.e even 9 1
108.2.l.a ✓ 96 36.h even 6 1
108.2.l.a ✓ 96 108.j odd 18 1
324.2.l.a 96 9.c even 3 1
324.2.l.a 96 27.f odd 18 1
324.2.l.a 96 36.f odd 6 1
324.2.l.a 96 108.l even 18 1
972.2.l.a 96 1.a even 1 1 trivial
972.2.l.a 96 4.b odd 2 1 inner
972.2.l.a 96 27.f odd 18 1 inner
972.2.l.a 96 108.l even 18 1 inner
972.2.l.b 96 9.c even 3 1
972.2.l.b 96 27.f odd 18 1
972.2.l.b 96 36.f odd 6 1
972.2.l.b 96 108.l even 18 1
972.2.l.c 96 9.d odd 6 1
972.2.l.c 96 27.e even 9 1
972.2.l.c 96 36.h even 6 1
972.2.l.c 96 108.j odd 18 1
972.2.l.d 96 3.b odd 2 1
972.2.l.d 96 12.b even 2 1
972.2.l.d 96 27.e even 9 1
972.2.l.d 96 108.j odd 18 1