Properties

Label 198.2.f.c
Level $198$
Weight $2$
Character orbit 198.f
Analytic conductor $1.581$
Analytic rank $0$
Dimension $4$
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(37,198)]
 
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([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 198.f (of order \(5\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{3} q^{4} + ( - 3 \zeta_{10}^{2} + \zeta_{10} - 3) q^{5} + ( - 3 \zeta_{10}^{3} + \zeta_{10} - 1) q^{7} - \zeta_{10}^{2} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{3} q^{4} + ( - 3 \zeta_{10}^{2} + \zeta_{10} - 3) q^{5} + ( - 3 \zeta_{10}^{3} + \zeta_{10} - 1) q^{7} - \zeta_{10}^{2} q^{8} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 2) q^{10} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10} + 1) q^{11} + (2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{13} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10}) q^{14} - \zeta_{10} q^{16} + (4 \zeta_{10}^{2} + 4) q^{17} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{19} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 2) q^{20} + ( - \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} + 3) q^{22} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{23} + (3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 3 \zeta_{10}) q^{25} + (2 \zeta_{10} - 2) q^{26} + (\zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{28} + ( - 2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{29} + (7 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 7) q^{31} - q^{32} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 4) q^{34} + (3 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} - 3) q^{35} + ( - 8 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{37} + (2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{38} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{40} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{41} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{43} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{44} + (2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{46} + (4 \zeta_{10}^{3} + 4 \zeta_{10}) q^{47} + (7 \zeta_{10}^{2} - 10 \zeta_{10} + 7) q^{49} + (3 \zeta_{10}^{2} + 5 \zeta_{10} + 3) q^{50} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{52} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{53} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 7 \zeta_{10} - 7) q^{55} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 3) q^{56} + ( - \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10}) q^{58} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{59} + (2 \zeta_{10}^{2} + 8 \zeta_{10} + 2) q^{61} + (7 \zeta_{10}^{3} - \zeta_{10} + 1) q^{62} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 6) q^{65} + 4 q^{67} + ( - 4 \zeta_{10}^{3} + 4) q^{68} + (3 \zeta_{10}^{3} + 10 \zeta_{10} - 10) q^{70} + ( - 4 \zeta_{10}^{2} + 10 \zeta_{10} - 4) q^{71} + ( - 7 \zeta_{10} + 7) q^{73} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{74} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{76} + ( - 9 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 15 \zeta_{10} + 7) q^{77} + (2 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} - 2) q^{79} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} + 3 \zeta_{10}) q^{80} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{82} + ( - 7 \zeta_{10}^{2} - 5 \zeta_{10} - 7) q^{83} + ( - 8 \zeta_{10}^{3} - 12 \zeta_{10}^{2} - 8 \zeta_{10}) q^{85} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{86} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} - 2) q^{88} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2) q^{89} + (8 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 8 \zeta_{10}) q^{91} + (2 \zeta_{10} - 2) q^{92} + (4 \zeta_{10}^{2} + 4) q^{94} + ( - 6 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{95} + ( - 7 \zeta_{10}^{2} + 7 \zeta_{10}) q^{97} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{4} - 8 q^{5} - 6 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{4} - 8 q^{5} - 6 q^{7} + q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 6 q^{14} - q^{16} + 12 q^{17} + 6 q^{19} - 8 q^{20} + 11 q^{22} + 4 q^{23} + q^{25} - 6 q^{26} - q^{28} + q^{29} - 5 q^{31} - 4 q^{32} + 8 q^{34} - 23 q^{35} + 4 q^{37} + 4 q^{38} - 7 q^{40} - 6 q^{41} - 4 q^{43} - q^{44} - 4 q^{46} + 8 q^{47} + 11 q^{49} + 14 q^{50} + 6 q^{52} + 7 q^{53} - 38 q^{55} - 14 q^{56} - q^{58} - 8 q^{59} + 14 q^{61} + 10 q^{62} - q^{64} + 28 q^{65} + 16 q^{67} + 12 q^{68} - 27 q^{70} - 2 q^{71} + 21 q^{73} - 4 q^{74} - 4 q^{76} - 6 q^{77} - 16 q^{79} + 7 q^{80} - 4 q^{82} - 26 q^{83} - 4 q^{85} + 4 q^{86} - 9 q^{88} + 20 q^{89} + 26 q^{91} - 6 q^{92} + 12 q^{94} - 12 q^{95} + 14 q^{97} - 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(145\) \(155\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i 0 0.309017 + 0.951057i −3.11803 + 2.26538i 0 0.736068 + 2.26538i −0.309017 + 0.951057i 0 −3.85410
91.1 0.809017 0.587785i 0 0.309017 0.951057i −3.11803 2.26538i 0 0.736068 2.26538i −0.309017 0.951057i 0 −3.85410
163.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i −0.881966 2.71441i 0 −3.73607 2.71441i 0.809017 0.587785i 0 2.85410
181.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i −0.881966 + 2.71441i 0 −3.73607 + 2.71441i 0.809017 + 0.587785i 0 2.85410
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.2.f.c 4
3.b odd 2 1 66.2.e.a 4
11.c even 5 1 inner 198.2.f.c 4
11.c even 5 1 2178.2.a.t 2
11.d odd 10 1 2178.2.a.bb 2
12.b even 2 1 528.2.y.d 4
33.d even 2 1 726.2.e.r 4
33.f even 10 1 726.2.a.j 2
33.f even 10 2 726.2.e.n 4
33.f even 10 1 726.2.e.r 4
33.h odd 10 1 66.2.e.a 4
33.h odd 10 1 726.2.a.l 2
33.h odd 10 2 726.2.e.f 4
132.n odd 10 1 5808.2.a.cg 2
132.o even 10 1 528.2.y.d 4
132.o even 10 1 5808.2.a.cb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.a 4 3.b odd 2 1
66.2.e.a 4 33.h odd 10 1
198.2.f.c 4 1.a even 1 1 trivial
198.2.f.c 4 11.c even 5 1 inner
528.2.y.d 4 12.b even 2 1
528.2.y.d 4 132.o even 10 1
726.2.a.j 2 33.f even 10 1
726.2.a.l 2 33.h odd 10 1
726.2.e.f 4 33.h odd 10 2
726.2.e.n 4 33.f even 10 2
726.2.e.r 4 33.d even 2 1
726.2.e.r 4 33.f even 10 1
2178.2.a.t 2 11.c even 5 1
2178.2.a.bb 2 11.d odd 10 1
5808.2.a.cb 2 132.o even 10 1
5808.2.a.cg 2 132.n odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(198, [\chi])\):

\( T_{5}^{4} + 8T_{5}^{3} + 34T_{5}^{2} + 77T_{5} + 121 \) Copy content Toggle raw display
\( T_{7}^{4} + 6T_{7}^{3} + 16T_{7}^{2} + 11T_{7} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121 \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 6 T^{2} - 44 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 12 T^{3} + 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - T^{3} + 6 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + 60 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 96 T^{2} + 256 T + 256 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256 \) Copy content Toggle raw display
$53$ \( T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121 \) Copy content Toggle raw display
$61$ \( T^{4} - 14 T^{3} + 96 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$67$ \( (T - 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + 124 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$73$ \( T^{4} - 21 T^{3} + 196 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( T^{4} + 16 T^{3} + 186 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$83$ \( T^{4} + 26 T^{3} + 256 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T - 20)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 14 T^{3} + 196 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
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