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Properties

Label 198.6.l.b
Level $198$
Weight $6$
Character orbit 198.l
Analytic conductor $31.756$
Analytic rank $0$
Dimension $40$
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] = [198,6,Mod(17,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 198.l (of order \(10\), degree \(4\), 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: \(31.7559963230\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{2} - 160 q^{4} + 640 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{2} - 160 q^{4} + 640 q^{8} - 476 q^{11} - 2560 q^{16} - 1424 q^{17} - 1656 q^{22} + 11574 q^{25} + 10480 q^{26} + 3040 q^{28} - 10658 q^{29} - 5302 q^{31} - 40960 q^{32} - 13984 q^{34} - 50 q^{35} - 4344 q^{37} + 35120 q^{38} + 19840 q^{40} - 16856 q^{41} + 6624 q^{44} - 52400 q^{46} - 10900 q^{47} - 40698 q^{49} - 6256 q^{50} + 41920 q^{52} + 10290 q^{53} + 158396 q^{55} + 42632 q^{58} + 62620 q^{59} - 134780 q^{61} - 30272 q^{62} - 40960 q^{64} - 137296 q^{65} - 36856 q^{67} - 22784 q^{68} + 79400 q^{70} + 62180 q^{71} + 100030 q^{73} + 17376 q^{74} - 162888 q^{77} + 35900 q^{79} + 79360 q^{80} - 90096 q^{82} - 12276 q^{83} + 270600 q^{85} - 137680 q^{86} - 26496 q^{88} + 406656 q^{91} + 134720 q^{92} - 91600 q^{94} - 75128 q^{95} - 364164 q^{97} + 358192 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i −68.3192 − 22.1982i 0 −35.8547 − 49.3497i 51.7771 + 37.6183i 0 287.340i
17.2 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i −67.8638 − 22.0503i 0 −151.878 − 209.042i 51.7771 + 37.6183i 0 285.425i
17.3 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i −55.4865 − 18.0286i 0 111.028 + 152.817i 51.7771 + 37.6183i 0 233.368i
17.4 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i −27.5399 − 8.94827i 0 −22.3921 − 30.8201i 51.7771 + 37.6183i 0 115.829i
17.5 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i −11.5390 − 3.74924i 0 −64.3821 − 88.6144i 51.7771 + 37.6183i 0 48.5312i
17.6 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i 19.8828 + 6.46032i 0 81.0836 + 111.602i 51.7771 + 37.6183i 0 − 83.6241i
17.7 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i 28.7854 + 9.35295i 0 56.7383 + 78.0936i 51.7771 + 37.6183i 0 − 121.067i
17.8 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i 71.9625 + 23.3820i 0 120.488 + 165.838i 51.7771 + 37.6183i 0 − 302.663i
17.9 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i 74.1025 + 24.0774i 0 −84.1988 − 115.890i 51.7771 + 37.6183i 0 − 311.664i
17.10 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i 105.333 + 34.2249i 0 −53.1179 − 73.1105i 51.7771 + 37.6183i 0 − 443.016i
35.1 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i −68.3192 + 22.1982i 0 −35.8547 + 49.3497i 51.7771 − 37.6183i 0 − 287.340i
35.2 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i −67.8638 + 22.0503i 0 −151.878 + 209.042i 51.7771 − 37.6183i 0 − 285.425i
35.3 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i −55.4865 + 18.0286i 0 111.028 − 152.817i 51.7771 − 37.6183i 0 − 233.368i
35.4 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i −27.5399 + 8.94827i 0 −22.3921 + 30.8201i 51.7771 − 37.6183i 0 − 115.829i
35.5 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i −11.5390 + 3.74924i 0 −64.3821 + 88.6144i 51.7771 − 37.6183i 0 − 48.5312i
35.6 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i 19.8828 − 6.46032i 0 81.0836 − 111.602i 51.7771 − 37.6183i 0 83.6241i
35.7 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i 28.7854 − 9.35295i 0 56.7383 − 78.0936i 51.7771 − 37.6183i 0 121.067i
35.8 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i 71.9625 − 23.3820i 0 120.488 − 165.838i 51.7771 − 37.6183i 0 302.663i
35.9 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i 74.1025 − 24.0774i 0 −84.1988 + 115.890i 51.7771 − 37.6183i 0 311.664i
35.10 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i 105.333 − 34.2249i 0 −53.1179 + 73.1105i 51.7771 − 37.6183i 0 443.016i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.6.l.b yes 40
3.b odd 2 1 198.6.l.a ✓ 40
11.d odd 10 1 198.6.l.a ✓ 40
33.f even 10 1 inner 198.6.l.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.6.l.a ✓ 40 3.b odd 2 1
198.6.l.a ✓ 40 11.d odd 10 1
198.6.l.b yes 40 1.a even 1 1 trivial
198.6.l.b yes 40 33.f even 10 1 inner