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Properties

Label 198.2.n.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{2} - 2 q^{3} + 6 q^{4} - 3 q^{5} + 6 q^{6} - 12 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{2} - 2 q^{3} + 6 q^{4} - 3 q^{5} + 6 q^{6} - 12 q^{8} - 20 q^{9} - 6 q^{11} + 6 q^{12} + 12 q^{13} - 27 q^{15} + 6 q^{16} - 6 q^{17} - 5 q^{18} + 15 q^{19} + 3 q^{20} + 9 q^{22} + 18 q^{23} - 2 q^{24} + 51 q^{25} - 32 q^{27} - 36 q^{29} - 24 q^{30} + 3 q^{31} - 24 q^{32} - 26 q^{33} + 3 q^{34} + 18 q^{35} - 10 q^{36} - 6 q^{37} - 30 q^{38} - 46 q^{39} - 3 q^{40} - 3 q^{41} + 8 q^{42} - 27 q^{43} + 12 q^{44} - 10 q^{45} + 3 q^{47} + q^{48} + 54 q^{49} - 9 q^{50} + 24 q^{51} + 18 q^{52} + 15 q^{54} + 6 q^{55} - 63 q^{57} - 36 q^{58} - 66 q^{59} - 4 q^{60} - 6 q^{62} - 6 q^{63} - 12 q^{64} + 69 q^{66} - 15 q^{67} + 3 q^{68} + 18 q^{69} + 6 q^{70} + 3 q^{74} + 14 q^{75} - 15 q^{76} - 18 q^{77} - 40 q^{78} - 42 q^{79} + 68 q^{81} - 9 q^{82} + 18 q^{83} + 22 q^{84} - 111 q^{85} + 27 q^{86} - 42 q^{87} - 6 q^{88} + 121 q^{90} - 36 q^{91} + 27 q^{92} + 135 q^{93} - 3 q^{94} + 132 q^{95} + q^{96} + 45 q^{97} - 48 q^{98} - 102 q^{99}+O(q^{100}) \) 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 −1.72891 + 0.104231i 0.669131 − 0.743145i 0.381381 − 0.856595i −1.53704 + 0.798432i 0.509520 − 2.39710i 0.309017 − 0.951057i 2.97827 − 0.360412i − 0.937660i
29.2 0.913545 − 0.406737i −0.453057 − 1.67175i 0.669131 − 0.743145i 0.615427 − 1.38227i −1.09385 − 1.34294i −0.0326192 + 0.153461i 0.309017 − 0.951057i −2.58948 + 1.51479i − 1.51309i
29.3 0.913545 − 0.406737i −0.313264 + 1.70349i 0.669131 − 0.743145i −0.963646 + 2.16439i 0.406689 + 1.68363i −0.748382 + 3.52086i 0.309017 − 0.951057i −2.80373 − 1.06728i 2.36921i
29.4 0.913545 − 0.406737i 0.639148 + 1.60981i 0.669131 − 0.743145i 0.778576 − 1.74871i 1.23866 + 1.21067i 0.413986 − 1.94765i 0.309017 − 0.951057i −2.18298 + 2.05781i − 1.91420i
29.5 0.913545 − 0.406737i 1.22686 − 1.22262i 0.669131 − 0.743145i −0.0289535 + 0.0650307i 0.623511 − 1.61593i −0.671482 + 3.15907i 0.309017 − 0.951057i 0.0103907 − 2.99998i 0.0711850i
29.6 0.913545 − 0.406737i 1.71190 + 0.263452i 0.669131 − 0.743145i −1.48727 + 3.34047i 1.67105 − 0.455616i 0.528978 − 2.48865i 0.309017 − 0.951057i 2.86119 + 0.902004i 3.65660i
41.1 0.913545 + 0.406737i −1.72891 − 0.104231i 0.669131 + 0.743145i 0.381381 + 0.856595i −1.53704 − 0.798432i 0.509520 + 2.39710i 0.309017 + 0.951057i 2.97827 + 0.360412i 0.937660i
41.2 0.913545 + 0.406737i −0.453057 + 1.67175i 0.669131 + 0.743145i 0.615427 + 1.38227i −1.09385 + 1.34294i −0.0326192 − 0.153461i 0.309017 + 0.951057i −2.58948 − 1.51479i 1.51309i
41.3 0.913545 + 0.406737i −0.313264 − 1.70349i 0.669131 + 0.743145i −0.963646 − 2.16439i 0.406689 − 1.68363i −0.748382 − 3.52086i 0.309017 + 0.951057i −2.80373 + 1.06728i − 2.36921i
41.4 0.913545 + 0.406737i 0.639148 − 1.60981i 0.669131 + 0.743145i 0.778576 + 1.74871i 1.23866 − 1.21067i 0.413986 + 1.94765i 0.309017 + 0.951057i −2.18298 − 2.05781i 1.91420i
41.5 0.913545 + 0.406737i 1.22686 + 1.22262i 0.669131 + 0.743145i −0.0289535 − 0.0650307i 0.623511 + 1.61593i −0.671482 − 3.15907i 0.309017 + 0.951057i 0.0103907 + 2.99998i − 0.0711850i
41.6 0.913545 + 0.406737i 1.71190 − 0.263452i 0.669131 + 0.743145i −1.48727 − 3.34047i 1.67105 + 0.455616i 0.528978 + 2.48865i 0.309017 + 0.951057i 2.86119 − 0.902004i − 3.65660i
83.1 −0.978148 − 0.207912i −1.69794 + 0.342076i 0.913545 + 0.406737i −0.103984 − 0.489207i 1.73195 + 0.0184197i −0.610544 + 0.0641707i −0.809017 − 0.587785i 2.76597 − 1.16165i 0.500136i
83.2 −0.978148 − 0.207912i −0.834906 − 1.51754i 0.913545 + 0.406737i 0.650698 + 3.06129i 0.501146 + 1.65797i −4.77384 + 0.501751i −0.809017 − 0.587785i −1.60587 + 2.53401i − 3.12969i
83.3 −0.978148 − 0.207912i −0.783089 − 1.54492i 0.913545 + 0.406737i −0.251484 − 1.18314i 0.444770 + 1.67397i 3.78966 − 0.398310i −0.809017 − 0.587785i −1.77354 + 2.41962i 1.20957i
83.4 −0.978148 − 0.207912i 0.103661 + 1.72895i 0.913545 + 0.406737i −0.322087 − 1.51530i 0.258072 − 1.71272i 3.90823 − 0.410772i −0.809017 − 0.587785i −2.97851 + 0.358450i 1.54915i
83.5 −0.978148 − 0.207912i 1.03825 − 1.38637i 0.913545 + 0.406737i −0.748462 − 3.52124i −1.30381 + 1.14021i −2.74349 + 0.288353i −0.809017 − 0.587785i −0.844066 − 2.87881i 3.59990i
83.6 −0.978148 − 0.207912i 1.60941 + 0.640145i 0.913545 + 0.406737i 0.415205 + 1.95338i −1.44115 − 0.960772i 0.429976 − 0.0451923i −0.809017 − 0.587785i 2.18043 + 2.06052i − 1.99702i
95.1 −0.104528 + 0.994522i −1.72659 − 0.137381i −0.978148 − 0.207912i −0.0180410 + 0.00189619i 0.317107 − 1.70278i −2.63326 − 2.37100i 0.309017 − 0.951057i 2.96225 + 0.474402i − 0.0181404i
95.2 −0.104528 + 0.994522i −1.36778 − 1.06263i −0.978148 − 0.207912i −0.289559 + 0.0304338i 1.19978 − 1.24921i 2.95795 + 2.66335i 0.309017 − 0.951057i 0.741631 + 2.90689i − 0.291154i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.6
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 3 T_{5}^{47} - 6 T_{5}^{46} - 27 T_{5}^{45} - 78 T_{5}^{44} + 654 T_{5}^{43} + \cdots + 891136 \) acting on \(S_{2}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display