Properties

Label 206.2.c.a
Level $206$
Weight $2$
Character orbit 206.c
Analytic conductor $1.645$
Analytic rank $0$
Dimension $2$
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] = [206,2,Mod(149,206)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(206, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("206.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 206 = 2 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 206.c (of order \(3\), degree \(2\), 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: \(1.64491828164\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + 2 q^{3} + (\zeta_{6} - 1) q^{4} + ( - 2 \zeta_{6} + 2) q^{5} + 2 \zeta_{6} q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + 2 q^{3} + (\zeta_{6} - 1) q^{4} + ( - 2 \zeta_{6} + 2) q^{5} + 2 \zeta_{6} q^{6} - q^{8} + q^{9} + 2 q^{10} + (2 \zeta_{6} - 2) q^{12} - 2 q^{13} + ( - 4 \zeta_{6} + 4) q^{15} - \zeta_{6} q^{16} + (5 \zeta_{6} - 5) q^{17} + \zeta_{6} q^{18} - 2 \zeta_{6} q^{19} + 2 \zeta_{6} q^{20} - 3 q^{23} - 2 q^{24} + \zeta_{6} q^{25} - 2 \zeta_{6} q^{26} - 4 q^{27} + 4 q^{30} + 5 q^{31} + ( - \zeta_{6} + 1) q^{32} - 5 q^{34} + (\zeta_{6} - 1) q^{36} + 2 q^{37} + ( - 2 \zeta_{6} + 2) q^{38} - 4 q^{39} + (2 \zeta_{6} - 2) q^{40} - 5 \zeta_{6} q^{41} - 2 \zeta_{6} q^{43} + ( - 2 \zeta_{6} + 2) q^{45} - 3 \zeta_{6} q^{46} + ( - 5 \zeta_{6} + 5) q^{47} - 2 \zeta_{6} q^{48} + 7 \zeta_{6} q^{49} + (\zeta_{6} - 1) q^{50} + (10 \zeta_{6} - 10) q^{51} + ( - 2 \zeta_{6} + 2) q^{52} + ( - 12 \zeta_{6} + 12) q^{53} - 4 \zeta_{6} q^{54} - 4 \zeta_{6} q^{57} - 6 \zeta_{6} q^{59} + 4 \zeta_{6} q^{60} - 2 q^{61} + 5 \zeta_{6} q^{62} + q^{64} + (4 \zeta_{6} - 4) q^{65} + (10 \zeta_{6} - 10) q^{67} - 5 \zeta_{6} q^{68} - 6 q^{69} + ( - 3 \zeta_{6} + 3) q^{71} - q^{72} + 7 q^{73} + 2 \zeta_{6} q^{74} + 2 \zeta_{6} q^{75} + 2 q^{76} - 4 \zeta_{6} q^{78} - 3 q^{79} - 2 q^{80} - 11 q^{81} + ( - 5 \zeta_{6} + 5) q^{82} - 14 \zeta_{6} q^{83} + 10 \zeta_{6} q^{85} + ( - 2 \zeta_{6} + 2) q^{86} + 11 q^{89} + 2 q^{90} + ( - 3 \zeta_{6} + 3) q^{92} + 10 q^{93} + 5 q^{94} - 4 q^{95} + ( - 2 \zeta_{6} + 2) q^{96} + 19 \zeta_{6} q^{97} + (7 \zeta_{6} - 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 4 q^{3} - q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 4 q^{3} - q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 4 q^{10} - 2 q^{12} - 4 q^{13} + 4 q^{15} - q^{16} - 5 q^{17} + q^{18} - 2 q^{19} + 2 q^{20} - 6 q^{23} - 4 q^{24} + q^{25} - 2 q^{26} - 8 q^{27} + 8 q^{30} + 10 q^{31} + q^{32} - 10 q^{34} - q^{36} + 4 q^{37} + 2 q^{38} - 8 q^{39} - 2 q^{40} - 5 q^{41} - 2 q^{43} + 2 q^{45} - 3 q^{46} + 5 q^{47} - 2 q^{48} + 7 q^{49} - q^{50} - 10 q^{51} + 2 q^{52} + 12 q^{53} - 4 q^{54} - 4 q^{57} - 6 q^{59} + 4 q^{60} - 4 q^{61} + 5 q^{62} + 2 q^{64} - 4 q^{65} - 10 q^{67} - 5 q^{68} - 12 q^{69} + 3 q^{71} - 2 q^{72} + 14 q^{73} + 2 q^{74} + 2 q^{75} + 4 q^{76} - 4 q^{78} - 6 q^{79} - 4 q^{80} - 22 q^{81} + 5 q^{82} - 14 q^{83} + 10 q^{85} + 2 q^{86} + 22 q^{89} + 4 q^{90} + 3 q^{92} + 20 q^{93} + 10 q^{94} - 8 q^{95} + 2 q^{96} + 19 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/206\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 2.00000 −0.500000 + 0.866025i 1.00000 1.73205i 1.00000 + 1.73205i 0 −1.00000 1.00000 2.00000
159.1 0.500000 0.866025i 2.00000 −0.500000 0.866025i 1.00000 + 1.73205i 1.00000 1.73205i 0 −1.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 206.2.c.a 2
3.b odd 2 1 1854.2.f.b 2
103.c even 3 1 inner 206.2.c.a 2
309.h odd 6 1 1854.2.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
206.2.c.a 2 1.a even 1 1 trivial
206.2.c.a 2 103.c even 3 1 inner
1854.2.f.b 2 3.b odd 2 1
1854.2.f.b 2 309.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(206, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( (T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$73$ \( (T - 7)^{2} \) Copy content Toggle raw display
$79$ \( (T + 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$89$ \( (T - 11)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
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