Properties

Label 198.2.l.a
Level $198$
Weight $2$
Character orbit 198.l
Analytic conductor $1.581$
Analytic rank $0$
Dimension $8$
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(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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.58103796002\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4} + \beta_{2} - 1) q^{2} - \beta_{6} q^{4} + (2 \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 - 1) q^{7} + \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{4} + \beta_{2} - 1) q^{2} - \beta_{6} q^{4} + (2 \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 - 1) q^{7} + \beta_{4} q^{8} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3}) q^{10} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{11} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{13} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{14} - \beta_{2} q^{16} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{2} + \beta_1) q^{19} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{20} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{22} + ( - \beta_{5} + 4 \beta_{4} - 4 \beta_{2} + 2) q^{23} + (2 \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + 2 \beta_1) q^{25} + ( - \beta_{7} + \beta_{5} - 2 \beta_{2} - 2) q^{26} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 1) q^{28} + ( - \beta_{7} - 4 \beta_{6} - \beta_{5} - 3 \beta_{2} + 3) q^{29} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 1) q^{31} + q^{32} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_1) q^{34} + (4 \beta_{7} - 5 \beta_{6} - \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 5) q^{35} + ( - 3 \beta_{7} - \beta_{5} - 2 \beta_{3} + \beta_1) q^{37} + (\beta_{7} - 2 \beta_{4} + \beta_{3} - \beta_1 + 2) q^{38} + ( - \beta_{7} + \beta_{5} - \beta_{2} - 1) q^{40} + ( - 3 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{41} + ( - \beta_{7} - 3 \beta_{5} + 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + 2) q^{43} + ( - \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_1) q^{44} + (2 \beta_{6} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2) q^{46} + ( - 4 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 4) q^{47} + ( - 2 \beta_{7} + 4 \beta_{5} - \beta_{4} - 6 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 1) q^{49} + (2 \beta_{7} - 4 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{50} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 4) q^{52} + (3 \beta_{6} - 3 \beta_{5} - 2 \beta_{3} - 3 \beta_1 + 3) q^{53} + (5 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{55} + (2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 1) q^{56} + (3 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2}) q^{58} + (3 \beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{2} + 2 \beta_1 - 3) q^{59} + (2 \beta_{7} - 8 \beta_{6} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 6) q^{61} + ( - 4 \beta_{7} - \beta_{6} + 2 \beta_{5} - 6 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 3) q^{62} + (\beta_{6} - \beta_{4} + \beta_{2} - 1) q^{64} + ( - 3 \beta_{7} + 3 \beta_{3} - 8) q^{65} + (2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 6 \beta_1 - 8) q^{67} + ( - 2 \beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{68} + ( - 3 \beta_{7} + 5 \beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_1) q^{70} + (3 \beta_{7} + 8 \beta_{6} - 6 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 2) q^{71} + ( - 4 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} - 3 \beta_{2} + 8 \beta_1 + 1) q^{73} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{74} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} - 2) q^{76} + (\beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} + 5 \beta_{2} - \beta_1 + 1) q^{77} + ( - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} + 8 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{79} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{80} + ( - 3 \beta_{7} + 6 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_1 + 2) q^{82} + ( - 3 \beta_{7} + 6 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 3) q^{83} + (5 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} + 6 \beta_{3} + 4 \beta_{2} - 5 \beta_1) q^{85} + (2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{86} + (\beta_{7} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{88} + (6 \beta_{6} - 6 \beta_{4} + 12 \beta_{2} - 6) q^{89} + (3 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{91} + (2 \beta_{6} - 4 \beta_{4} + 4 \beta_{2} - \beta_1) q^{92} + (4 \beta_{6} + \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{94} + (2 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{95} + (8 \beta_{6} - 2 \beta_{5} - 9 \beta_{4} + 9 \beta_{2} + 2 \beta_1 - 8) q^{97} + ( - 2 \beta_{7} - \beta_{6} + 4 \beta_{5} + \beta_{4} - 6 \beta_{3} + 8 \beta_1 - 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 4 q^{11} - 2 q^{16} - 8 q^{17} - 6 q^{22} + 6 q^{25} - 20 q^{26} - 10 q^{28} + 10 q^{29} - 14 q^{31} + 8 q^{32} - 8 q^{34} + 10 q^{35} + 20 q^{38} - 10 q^{40} - 8 q^{41} - 6 q^{44} + 20 q^{46} + 20 q^{47} + 6 q^{49} - 4 q^{50} + 20 q^{52} + 30 q^{53} + 28 q^{55} + 10 q^{58} - 20 q^{59} + 20 q^{61} + 16 q^{62} - 2 q^{64} - 64 q^{65} - 56 q^{67} - 8 q^{68} + 10 q^{70} + 20 q^{71} - 10 q^{73} - 20 q^{79} + 10 q^{80} + 12 q^{82} - 12 q^{83} + 20 q^{86} - 6 q^{88} + 20 q^{92} - 20 q^{94} + 16 q^{95} - 12 q^{97} - 44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/198\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(155\)
\(\chi(n)\) \(\beta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.831254 1.14412i
0.831254 + 1.14412i
−0.831254 + 1.14412i
0.831254 1.14412i
−1.34500 0.437016i
1.34500 + 0.437016i
−1.34500 + 0.437016i
1.34500 0.437016i
0.309017 + 0.951057i 0 −0.809017 + 0.587785i −1.05822 0.343836i 0 2.97677 + 4.09718i −0.809017 0.587785i 0 1.11268i
17.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 3.29428 + 1.07038i 0 −0.740706 1.01949i −0.809017 0.587785i 0 3.46382i
35.1 0.309017 0.951057i 0 −0.809017 0.587785i −1.05822 + 0.343836i 0 2.97677 4.09718i −0.809017 + 0.587785i 0 1.11268i
35.2 0.309017 0.951057i 0 −0.809017 0.587785i 3.29428 1.07038i 0 −0.740706 + 1.01949i −0.809017 + 0.587785i 0 3.46382i
107.1 −0.809017 + 0.587785i 0 0.309017 0.951057i −1.63178 + 2.24595i 0 −4.12554 1.34047i 0.309017 + 0.951057i 0 2.77615i
107.2 −0.809017 + 0.587785i 0 0.309017 0.951057i −0.604291 + 0.831735i 0 1.88947 + 0.613926i 0.309017 + 0.951057i 0 1.02808i
161.1 −0.809017 0.587785i 0 0.309017 + 0.951057i −1.63178 2.24595i 0 −4.12554 + 1.34047i 0.309017 0.951057i 0 2.77615i
161.2 −0.809017 0.587785i 0 0.309017 + 0.951057i −0.604291 0.831735i 0 1.88947 0.613926i 0.309017 0.951057i 0 1.02808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.2
Significant digits:
Format:

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 198.2.l.a 8
3.b odd 2 1 198.2.l.b yes 8
4.b odd 2 1 1584.2.cd.a 8
11.c even 5 1 2178.2.b.j 8
11.d odd 10 1 198.2.l.b yes 8
11.d odd 10 1 2178.2.b.i 8
12.b even 2 1 1584.2.cd.b 8
33.f even 10 1 inner 198.2.l.a 8
33.f even 10 1 2178.2.b.j 8
33.h odd 10 1 2178.2.b.i 8
44.g even 10 1 1584.2.cd.b 8
132.n odd 10 1 1584.2.cd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.2.l.a 8 1.a even 1 1 trivial
198.2.l.a 8 33.f even 10 1 inner
198.2.l.b yes 8 3.b odd 2 1
198.2.l.b yes 8 11.d odd 10 1
1584.2.cd.a 8 4.b odd 2 1
1584.2.cd.a 8 132.n odd 10 1
1584.2.cd.b 8 12.b even 2 1
1584.2.cd.b 8 44.g even 10 1
2178.2.b.i 8 11.d odd 10 1
2178.2.b.i 8 33.h odd 10 1
2178.2.b.j 8 11.c even 5 1
2178.2.b.j 8 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 8T_{5}^{6} - 30T_{5}^{5} + 34T_{5}^{4} + 240T_{5}^{3} + 403T_{5}^{2} + 330T_{5} + 121 \) acting on \(S_{2}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{6} - 30 T^{5} + 34 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( T^{8} - 10 T^{6} + 110 T^{5} + \cdots + 3025 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + 44 T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{6} + 120 T^{5} + 184 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + 48 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{6} + 120 T^{5} + 184 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{8} + 88 T^{6} + 1944 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} - 10 T^{7} + 115 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$31$ \( T^{8} + 14 T^{7} + 147 T^{6} + \cdots + 1771561 \) Copy content Toggle raw display
$37$ \( T^{8} + 80 T^{6} + 2440 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} - 2 T^{6} + \cdots + 234256 \) Copy content Toggle raw display
$43$ \( T^{8} + 188 T^{6} + 11164 T^{4} + \cdots + 1157776 \) Copy content Toggle raw display
$47$ \( T^{8} - 20 T^{7} + 142 T^{6} + \cdots + 13456 \) Copy content Toggle raw display
$53$ \( T^{8} - 30 T^{7} + 463 T^{6} + \cdots + 7027801 \) Copy content Toggle raw display
$59$ \( T^{8} + 20 T^{7} + 218 T^{6} + \cdots + 5387041 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} + \cdots + 198697216 \) Copy content Toggle raw display
$67$ \( (T^{4} + 28 T^{3} + 184 T^{2} - 248 T - 1604)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + 118 T^{6} + \cdots + 1628176 \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{7} - 287 T^{6} + \cdots + 19456921 \) Copy content Toggle raw display
$79$ \( T^{8} + 20 T^{7} - 102 T^{6} + \cdots + 101761 \) Copy content Toggle raw display
$83$ \( T^{8} + 12 T^{7} + 138 T^{6} + \cdots + 942841 \) Copy content Toggle raw display
$89$ \( (T^{4} + 180 T^{2} + 6480)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 12 T^{7} + 228 T^{6} + \cdots + 7623121 \) Copy content Toggle raw display
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