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Properties

Label 198.2.n.a
Level $198$
Weight $2$
Character orbit 198.n
Analytic conductor $1.581$
Analytic rank $0$
Dimension $48$
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] = [198,2,Mod(29,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([5, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 198.n (of order \(30\), degree \(8\), 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: \(1.58103796002\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 3 q^{5} - q^{6} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 3 q^{5} - q^{6} + 12 q^{8} - 9 q^{11} + 6 q^{12} - 12 q^{13} + 33 q^{15} + 6 q^{16} + 6 q^{17} - 5 q^{18} + 15 q^{19} + 3 q^{20} - 6 q^{22} + 18 q^{23} - 3 q^{24} - 69 q^{25} - 2 q^{27} - 54 q^{29} - 6 q^{30} + 3 q^{31} + 24 q^{32} + 4 q^{33} + 3 q^{34} - 18 q^{35} - 5 q^{36} - 6 q^{37} - 30 q^{38} - 24 q^{39} + 3 q^{40} + 3 q^{41} - 52 q^{42} + 27 q^{43} - 12 q^{44} - 10 q^{45} + 3 q^{47} + q^{48} - 66 q^{49} + 9 q^{50} - 9 q^{51} - 18 q^{52} - 15 q^{54} + 6 q^{55} + 13 q^{57} + 54 q^{58} - 51 q^{59} - 4 q^{60} + 6 q^{62} - 4 q^{63} - 12 q^{64} - 29 q^{66} - 15 q^{67} - 3 q^{68} - 82 q^{69} + 6 q^{70} + 20 q^{72} - 3 q^{74} + 49 q^{75} + 15 q^{76} - 36 q^{77} - 40 q^{78} + 42 q^{79} + 28 q^{81} - 9 q^{82} + 72 q^{83} + 38 q^{84} + 111 q^{85} + 12 q^{86} + 42 q^{87} + 9 q^{88} + 19 q^{90} - 36 q^{91} + 27 q^{92} - 25 q^{93} + 3 q^{94} + 108 q^{95} - q^{96} - 60 q^{97} + 48 q^{98} + 104 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.913545 + 0.406737i −1.52094 + 0.828691i 0.669131 − 0.743145i −1.44871 + 3.25385i 1.05239 − 1.37567i 0.197937 − 0.931222i −0.309017 + 0.951057i 1.62654 − 2.52078i − 3.56178i
29.2 −0.913545 + 0.406737i −0.474682 + 1.66574i 0.669131 − 0.743145i 1.36124 − 3.05740i −0.243873 − 1.71480i 0.761873 − 3.58433i −0.309017 + 0.951057i −2.54935 − 1.58139i 3.34674i
29.3 −0.913545 + 0.406737i 0.115214 − 1.72821i 0.669131 − 0.743145i 1.26197 − 2.83444i 0.597675 + 1.62566i −0.407389 + 1.91662i −0.309017 + 0.951057i −2.97345 − 0.398228i 3.10268i
29.4 −0.913545 + 0.406737i 0.627959 − 1.61421i 0.669131 − 0.743145i −1.75320 + 3.93775i 0.0828883 + 1.73007i −0.643262 + 3.02631i −0.309017 + 0.951057i −2.21133 − 2.02731i − 4.31040i
29.5 −0.913545 + 0.406737i 1.47759 + 0.903723i 0.669131 − 0.743145i −0.00737836 + 0.0165721i −1.71743 − 0.224600i −0.736714 + 3.46597i −0.309017 + 0.951057i 1.36657 + 2.67067i − 0.0181404i
29.6 −0.913545 + 0.406737i 1.73115 − 0.0557271i 0.669131 − 0.743145i −0.118423 + 0.265982i −1.55882 + 0.755033i 0.827555 − 3.89334i −0.309017 + 0.951057i 2.99379 − 0.192945i − 0.291154i
41.1 −0.913545 − 0.406737i −1.52094 − 0.828691i 0.669131 + 0.743145i −1.44871 − 3.25385i 1.05239 + 1.37567i 0.197937 + 0.931222i −0.309017 − 0.951057i 1.62654 + 2.52078i 3.56178i
41.2 −0.913545 − 0.406737i −0.474682 − 1.66574i 0.669131 + 0.743145i 1.36124 + 3.05740i −0.243873 + 1.71480i 0.761873 + 3.58433i −0.309017 − 0.951057i −2.54935 + 1.58139i − 3.34674i
41.3 −0.913545 − 0.406737i 0.115214 + 1.72821i 0.669131 + 0.743145i 1.26197 + 2.83444i 0.597675 − 1.62566i −0.407389 − 1.91662i −0.309017 − 0.951057i −2.97345 + 0.398228i − 3.10268i
41.4 −0.913545 − 0.406737i 0.627959 + 1.61421i 0.669131 + 0.743145i −1.75320 − 3.93775i 0.0828883 − 1.73007i −0.643262 − 3.02631i −0.309017 − 0.951057i −2.21133 + 2.02731i 4.31040i
41.5 −0.913545 − 0.406737i 1.47759 − 0.903723i 0.669131 + 0.743145i −0.00737836 − 0.0165721i −1.71743 + 0.224600i −0.736714 − 3.46597i −0.309017 − 0.951057i 1.36657 − 2.67067i 0.0181404i
41.6 −0.913545 − 0.406737i 1.73115 + 0.0557271i 0.669131 + 0.743145i −0.118423 − 0.265982i −1.55882 − 0.755033i 0.827555 + 3.89334i −0.309017 − 0.951057i 2.99379 + 0.192945i 0.291154i
83.1 0.978148 + 0.207912i −1.65599 + 0.507641i 0.913545 + 0.406737i −0.869131 − 4.08894i −1.72535 + 0.152249i 4.49435 − 0.472376i 0.809017 + 0.587785i 2.48460 − 1.68130i − 4.18029i
83.2 0.978148 + 0.207912i −1.19280 + 1.25588i 0.913545 + 0.406737i 0.461700 + 2.17213i −1.42784 + 0.980440i −2.78045 + 0.292237i 0.809017 + 0.587785i −0.154472 − 2.99602i 2.22065i
83.3 0.978148 + 0.207912i −0.119381 − 1.72793i 0.913545 + 0.406737i −0.446758 − 2.10183i 0.242485 − 1.71499i −0.460263 + 0.0483756i 0.809017 + 0.587785i −2.97150 + 0.412565i − 2.14879i
83.4 0.978148 + 0.207912i 0.661337 + 1.60082i 0.913545 + 0.406737i −0.0500878 − 0.235645i 0.314055 + 1.70334i 0.613832 − 0.0645163i 0.809017 + 0.587785i −2.12527 + 2.11737i − 0.240909i
83.5 0.978148 + 0.207912i 0.786343 − 1.54326i 0.913545 + 0.406737i 0.871756 + 4.10129i 1.09002 − 1.34605i 0.511746 − 0.0537867i 0.809017 + 0.587785i −1.76333 − 2.42707i 4.19291i
83.6 0.978148 + 0.207912i 1.72954 − 0.0931485i 0.913545 + 0.406737i −0.327592 − 1.54120i 1.71112 + 0.268479i −2.37922 + 0.250066i 0.809017 + 0.587785i 2.98265 − 0.322209i − 1.57563i
95.1 0.104528 − 0.994522i −1.53981 − 0.793091i −0.978148 − 0.207912i −3.63657 + 0.382219i −0.949700 + 1.44847i 1.89074 + 1.70243i −0.309017 + 0.951057i 1.74201 + 2.44242i 3.65660i
95.2 0.104528 − 0.994522i −1.46330 + 0.926683i −0.978148 − 0.207912i 1.90372 − 0.200089i 0.768649 + 1.55215i 1.47972 + 1.33235i −0.309017 + 0.951057i 1.28252 − 2.71204i − 1.91420i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
99.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.2.n.a ✓ 48
3.b odd 2 1 594.2.r.b 48
9.c even 3 1 594.2.r.a 48
9.d odd 6 1 198.2.n.b yes 48
11.d odd 10 1 198.2.n.b yes 48
33.f even 10 1 594.2.r.a 48
99.o odd 30 1 594.2.r.b 48
99.p even 30 1 inner 198.2.n.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.2.n.a ✓ 48 1.a even 1 1 trivial
198.2.n.a ✓ 48 99.p even 30 1 inner
198.2.n.b yes 48 9.d odd 6 1
198.2.n.b yes 48 11.d odd 10 1
594.2.r.a 48 9.c even 3 1
594.2.r.a 48 33.f even 10 1
594.2.r.b 48 3.b odd 2 1
594.2.r.b 48 99.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 3 T_{5}^{47} + 54 T_{5}^{46} + 153 T_{5}^{45} + 1302 T_{5}^{44} + 4254 T_{5}^{43} + \cdots + 891136 \) acting on \(S_{2}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display