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Properties

Label 162.4.g.b
Level $162$
Weight $4$
Character orbit 162.g
Analytic conductor $9.558$
Analytic rank $0$
Dimension $252$
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] = [162,4,Mod(7,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(54))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.g (of order \(27\), degree \(18\), 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: \(9.55830942093\)
Analytic rank: \(0\)
Dimension: \(252\)
Relative dimension: \(14\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 252 q - 36 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 252 q - 36 q^{6} + 90 q^{13} - 252 q^{18} - 144 q^{20} + 189 q^{21} + 1512 q^{23} + 846 q^{25} + 702 q^{26} + 702 q^{27} - 504 q^{28} + 540 q^{29} - 342 q^{30} - 2214 q^{31} - 1548 q^{33} - 1242 q^{35} - 576 q^{36} - 882 q^{38} - 927 q^{41} + 774 q^{42} - 900 q^{43} + 2817 q^{45} - 2088 q^{46} + 297 q^{47} - 144 q^{48} + 2151 q^{51} - 720 q^{52} + 1431 q^{53} - 2970 q^{55} + 729 q^{57} + 126 q^{58} + 2628 q^{59} + 261 q^{63} + 3627 q^{65} + 4680 q^{66} + 2538 q^{67} - 936 q^{68} + 4716 q^{69} - 3150 q^{70} + 720 q^{71} - 1440 q^{72} + 3204 q^{73} - 2088 q^{74} + 918 q^{75} + 1764 q^{76} - 9792 q^{77} - 3132 q^{78} + 3897 q^{79} - 1440 q^{80} - 17352 q^{81} + 5904 q^{82} - 1035 q^{83} - 1224 q^{84} + 2421 q^{85} - 3600 q^{86} - 10827 q^{87} + 1224 q^{88} - 4086 q^{89} + 4473 q^{91} + 1872 q^{92} + 8955 q^{93} - 6174 q^{94} + 8037 q^{95} + 1152 q^{96} + 6192 q^{97} + 8136 q^{98} + 2187 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.19432 − 1.60425i −4.99896 − 1.41789i −1.14721 + 3.83196i 13.9580 − 9.18030i 3.69570 + 9.71297i −16.3937 + 17.3763i 7.51754 − 2.73616i 22.9792 + 14.1759i −31.3977 − 11.4278i
7.2 −1.19432 − 1.60425i −4.83160 + 1.91197i −1.14721 + 3.83196i 1.51049 − 0.993463i 8.83774 + 5.46758i 8.41336 − 8.91764i 7.51754 − 2.73616i 19.6887 − 18.4758i −3.39776 − 1.23668i
7.3 −1.19432 − 1.60425i −4.55952 − 2.49214i −1.14721 + 3.83196i −10.2500 + 6.74151i 1.44751 + 10.2910i −13.2497 + 14.0439i 7.51754 − 2.73616i 14.5785 + 22.7259i 23.0567 + 8.39197i
7.4 −1.19432 − 1.60425i −3.47518 − 3.86305i −1.14721 + 3.83196i 5.82437 − 3.83075i −2.04682 + 10.1887i 21.8767 − 23.1880i 7.51754 − 2.73616i −2.84630 + 26.8496i −13.1016 − 4.76860i
7.5 −1.19432 − 1.60425i −3.14794 + 4.13406i −1.14721 + 3.83196i −12.1611 + 7.99847i 10.3917 + 0.112698i −13.0673 + 13.8505i 7.51754 − 2.73616i −7.18091 − 26.0276i 27.3557 + 9.95667i
7.6 −1.19432 − 1.60425i −0.821241 − 5.13084i −1.14721 + 3.83196i 1.88361 − 1.23887i −7.25032 + 7.44533i −2.53643 + 2.68846i 7.51754 − 2.73616i −25.6511 + 8.42732i −4.23709 − 1.54217i
7.7 −1.19432 − 1.60425i −0.639106 + 5.15670i −1.14721 + 3.83196i 4.47381 − 2.94247i 9.03591 − 5.13345i 8.78882 − 9.31560i 7.51754 − 2.73616i −26.1831 − 6.59135i −10.0636 − 3.66285i
7.8 −1.19432 − 1.60425i 0.771405 − 5.13857i −1.14721 + 3.83196i −17.0853 + 11.2372i −9.16484 + 4.89956i 7.54659 − 7.99892i 7.51754 − 2.73616i −25.8099 − 7.92784i 38.4326 + 13.9883i
7.9 −1.19432 − 1.60425i 2.06068 + 4.77007i −1.14721 + 3.83196i 6.94326 − 4.56666i 5.19127 − 9.00282i −24.6415 + 26.1185i 7.51754 − 2.73616i −18.5072 + 19.6592i −15.6185 − 5.68467i
7.10 −1.19432 − 1.60425i 2.83197 + 4.35660i −1.14721 + 3.83196i −15.4172 + 10.1401i 3.60678 − 9.74634i 13.2031 − 13.9945i 7.51754 − 2.73616i −10.9599 + 24.6755i 34.6802 + 12.6226i
7.11 −1.19432 − 1.60425i 3.30686 − 4.00807i −1.14721 + 3.83196i 18.1458 − 11.9347i −10.3794 − 0.518112i −8.00040 + 8.47993i 7.51754 − 2.73616i −5.12931 − 26.5083i −40.8181 − 14.8566i
7.12 −1.19432 − 1.60425i 3.85407 − 3.48513i −1.14721 + 3.83196i −1.04141 + 0.684943i −10.1940 − 2.02053i 3.33832 − 3.53842i 7.51754 − 2.73616i 2.70774 − 26.8639i 2.34259 + 0.852632i
7.13 −1.19432 − 1.60425i 4.68723 + 2.24275i −1.14721 + 3.83196i 16.0370 − 10.5477i −2.00012 − 10.1980i 18.3946 − 19.4971i 7.51754 − 2.73616i 16.9402 + 21.0245i −36.0744 − 13.1300i
7.14 −1.19432 − 1.60425i 5.17909 − 0.420697i −1.14721 + 3.83196i −4.05716 + 2.66844i −6.86038 − 7.80610i −14.1577 + 15.0063i 7.51754 − 2.73616i 26.6460 − 4.35766i 9.12637 + 3.32173i
13.1 −1.78727 − 0.897598i −5.19032 + 0.246219i 2.38863 + 3.20849i 4.68865 + 15.6612i 9.49748 + 4.21876i −0.254938 − 0.591012i −1.38919 − 7.87846i 26.8788 − 2.55591i 5.67759 − 32.1992i
13.2 −1.78727 − 0.897598i −5.16958 − 0.524798i 2.38863 + 3.20849i −4.83465 − 16.1489i 8.76836 + 5.57816i −8.83771 − 20.4881i −1.38919 − 7.87846i 26.4492 + 5.42597i −5.85438 + 33.2019i
13.3 −1.78727 − 0.897598i −3.98094 − 3.33948i 2.38863 + 3.20849i −3.33686 − 11.1459i 4.11748 + 9.54182i 11.7915 + 27.3357i −1.38919 − 7.87846i 4.69572 + 26.5885i −4.04068 + 22.9158i
13.4 −1.78727 − 0.897598i −3.27564 + 4.03363i 2.38863 + 3.20849i −0.224791 − 0.750853i 9.47502 − 4.26895i −4.80994 − 11.1507i −1.38919 − 7.87846i −5.54032 − 26.4255i −0.272204 + 1.54374i
13.5 −1.78727 − 0.897598i −3.15164 − 4.13124i 2.38863 + 3.20849i 1.14961 + 3.83998i 1.92461 + 10.2125i −0.280834 − 0.651047i −1.38919 − 7.87846i −7.13437 + 26.0404i 1.39209 − 7.89495i
13.6 −1.78727 − 0.897598i −0.479474 + 5.17398i 2.38863 + 3.20849i −5.85860 − 19.5691i 5.50111 − 8.81691i 9.36681 + 21.7147i −1.38919 − 7.87846i −26.5402 − 4.96158i −7.09430 + 40.2338i
See next 80 embeddings (of 252 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.g even 27 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.4.g.b ✓ 252
81.g even 27 1 inner 162.4.g.b ✓ 252
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.g.b ✓ 252 1.a even 1 1 trivial
162.4.g.b ✓ 252 81.g even 27 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{252} - 423 T_{5}^{250} - 591 T_{5}^{249} + 102996 T_{5}^{248} + 2620053 T_{5}^{247} + \cdots + 10\!\cdots\!84 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display