Properties

Label 201.1.g.a
Level $201$
Weight $1$
Character orbit 201.g
Analytic conductor $0.100$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,1,Mod(29,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.29");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 201.g (of order \(6\), 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: \(0.100312067539\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.40401.1
Artin image: $\SL(2,3):C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} - \zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} - \zeta_{12}^{3} q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} - \zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} - \zeta_{12}^{3} q^{8} - q^{9} + \zeta_{12} q^{11} - \zeta_{12}^{2} q^{13} - \zeta_{12}^{3} q^{14} + \zeta_{12}^{2} q^{16} + \zeta_{12}^{5} q^{17} - \zeta_{12}^{5} q^{18} + \zeta_{12}^{2} q^{19} - \zeta_{12} q^{21} - q^{22} - \zeta_{12}^{5} q^{23} + q^{24} + q^{25} + \zeta_{12} q^{26} - \zeta_{12}^{3} q^{27} + \zeta_{12} q^{29} - \zeta_{12}^{4} q^{31} + \zeta_{12} q^{32} + \zeta_{12}^{4} q^{33} - \zeta_{12}^{4} q^{34} - \zeta_{12}^{2} q^{37} - \zeta_{12} q^{38} - \zeta_{12}^{5} q^{39} - \zeta_{12} q^{41} + q^{42} + \zeta_{12}^{4} q^{46} + \zeta_{12} q^{47} + \zeta_{12}^{5} q^{48} + \zeta_{12}^{5} q^{50} - \zeta_{12}^{2} q^{51} + \zeta_{12}^{2} q^{54} + \zeta_{12} q^{56} + \zeta_{12}^{5} q^{57} - q^{58} - \zeta_{12}^{2} q^{61} + \zeta_{12}^{3} q^{62} - \zeta_{12}^{4} q^{63} - q^{64} - \zeta_{12}^{3} q^{66} - q^{67} + \zeta_{12}^{2} q^{69} - \zeta_{12} q^{71} + \zeta_{12}^{3} q^{72} + \zeta_{12}^{2} q^{73} + \zeta_{12} q^{74} + \zeta_{12}^{3} q^{75} + \zeta_{12}^{5} q^{77} + \zeta_{12}^{4} q^{78} + \zeta_{12}^{4} q^{79} + q^{81} + q^{82} + \zeta_{12}^{5} q^{83} + \zeta_{12}^{4} q^{87} - \zeta_{12}^{4} q^{88} - \zeta_{12}^{3} q^{89} + q^{91} + \zeta_{12} q^{93} - q^{94} - \zeta_{12}^{2} q^{97} - \zeta_{12} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{6} - 2 q^{7} - 4 q^{9} - 2 q^{13} + 2 q^{16} + 2 q^{19} - 4 q^{22} + 4 q^{24} + 4 q^{25} + 2 q^{31} - 2 q^{33} + 2 q^{34} - 2 q^{37} + 4 q^{42} - 2 q^{46} - 2 q^{51} + 2 q^{54} - 4 q^{58} - 2 q^{61} + 2 q^{63} - 4 q^{64} - 4 q^{67} + 2 q^{69} + 2 q^{73} - 2 q^{78} - 2 q^{79} + 4 q^{81} + 4 q^{82} - 2 q^{87} + 2 q^{88} + 4 q^{91} - 4 q^{94} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i 1.00000i 0 0 −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i −1.00000 0
29.2 0.866025 + 0.500000i 1.00000i 0 0 −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i −1.00000 0
104.1 −0.866025 + 0.500000i 1.00000i 0 0 −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 0
104.2 0.866025 0.500000i 1.00000i 0 0 −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.c even 3 1 inner
201.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.1.g.a 4
3.b odd 2 1 inner 201.1.g.a 4
4.b odd 2 1 3216.1.bb.b 4
12.b even 2 1 3216.1.bb.b 4
67.c even 3 1 inner 201.1.g.a 4
201.g odd 6 1 inner 201.1.g.a 4
268.g odd 6 1 3216.1.bb.b 4
804.l even 6 1 3216.1.bb.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.1.g.a 4 1.a even 1 1 trivial
201.1.g.a 4 3.b odd 2 1 inner
201.1.g.a 4 67.c even 3 1 inner
201.1.g.a 4 201.g odd 6 1 inner
3216.1.bb.b 4 4.b odd 2 1
3216.1.bb.b 4 12.b even 2 1
3216.1.bb.b 4 268.g odd 6 1
3216.1.bb.b 4 804.l even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(201, [\chi])\).

Hecke characteristic polynomials

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