Properties

Label 133.2.i.a
Level $133$
Weight $2$
Character orbit 133.i
Analytic conductor $1.062$
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] = [133,2,Mod(12,133)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(133, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("133.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.i (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: \(1.06201034688\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - q^{4} + (\zeta_{6} - 2) q^{6} + ( - 2 \zeta_{6} - 1) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 2 \zeta_{6} + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - q^{4} + (\zeta_{6} - 2) q^{6} + ( - 2 \zeta_{6} - 1) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 2 \zeta_{6} + 2) q^{9} + (3 \zeta_{6} - 3) q^{11} + \zeta_{6} q^{12} + 5 \zeta_{6} q^{13} + (4 \zeta_{6} - 5) q^{14} - 5 q^{16} + ( - \zeta_{6} + 2) q^{17} + ( - 2 \zeta_{6} - 2) q^{18} + (2 \zeta_{6} + 3) q^{19} + (3 \zeta_{6} - 2) q^{21} + (3 \zeta_{6} + 3) q^{22} + ( - 9 \zeta_{6} + 9) q^{23} + (\zeta_{6} - 2) q^{24} + 5 q^{25} + ( - 5 \zeta_{6} + 10) q^{26} - 5 q^{27} + (2 \zeta_{6} + 1) q^{28} + ( - \zeta_{6} + 2) q^{29} + (7 \zeta_{6} - 7) q^{31} + (6 \zeta_{6} - 3) q^{32} + 3 q^{33} - 3 \zeta_{6} q^{34} + (2 \zeta_{6} - 2) q^{36} + ( - \zeta_{6} + 2) q^{37} + ( - 8 \zeta_{6} + 7) q^{38} + ( - 5 \zeta_{6} + 5) q^{39} + (3 \zeta_{6} - 3) q^{41} + (\zeta_{6} + 4) q^{42} + (7 \zeta_{6} - 7) q^{43} + ( - 3 \zeta_{6} + 3) q^{44} + ( - 9 \zeta_{6} - 9) q^{46} + (\zeta_{6} + 1) q^{47} + 5 \zeta_{6} q^{48} + (8 \zeta_{6} - 3) q^{49} + ( - 10 \zeta_{6} + 5) q^{50} + ( - \zeta_{6} - 1) q^{51} - 5 \zeta_{6} q^{52} + (10 \zeta_{6} - 5) q^{54} + (4 \zeta_{6} - 5) q^{56} + ( - 5 \zeta_{6} + 2) q^{57} - 3 \zeta_{6} q^{58} + 3 \zeta_{6} q^{59} + ( - 3 \zeta_{6} - 3) q^{61} + (7 \zeta_{6} + 7) q^{62} + (2 \zeta_{6} - 6) q^{63} - q^{64} + ( - 6 \zeta_{6} + 3) q^{66} + (12 \zeta_{6} - 6) q^{67} + (\zeta_{6} - 2) q^{68} - 9 q^{69} + ( - 7 \zeta_{6} - 7) q^{71} + ( - 2 \zeta_{6} - 2) q^{72} + ( - \zeta_{6} + 2) q^{73} - 3 \zeta_{6} q^{74} - 5 \zeta_{6} q^{75} + ( - 2 \zeta_{6} - 3) q^{76} + ( - 3 \zeta_{6} + 9) q^{77} + ( - 5 \zeta_{6} - 5) q^{78} + (4 \zeta_{6} - 2) q^{79} - \zeta_{6} q^{81} + (3 \zeta_{6} + 3) q^{82} + ( - 20 \zeta_{6} + 10) q^{83} + ( - 3 \zeta_{6} + 2) q^{84} + (7 \zeta_{6} + 7) q^{86} + ( - \zeta_{6} - 1) q^{87} + (3 \zeta_{6} + 3) q^{88} + (15 \zeta_{6} - 15) q^{89} + ( - 15 \zeta_{6} + 10) q^{91} + (9 \zeta_{6} - 9) q^{92} + 7 q^{93} + ( - 3 \zeta_{6} + 3) q^{94} + ( - 3 \zeta_{6} + 6) q^{96} + ( - \zeta_{6} + 1) q^{97} + ( - 2 \zeta_{6} + 13) q^{98} + 6 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{4} - 3 q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 2 q^{4} - 3 q^{6} - 4 q^{7} + 2 q^{9} - 3 q^{11} + q^{12} + 5 q^{13} - 6 q^{14} - 10 q^{16} + 3 q^{17} - 6 q^{18} + 8 q^{19} - q^{21} + 9 q^{22} + 9 q^{23} - 3 q^{24} + 10 q^{25} + 15 q^{26} - 10 q^{27} + 4 q^{28} + 3 q^{29} - 7 q^{31} + 6 q^{33} - 3 q^{34} - 2 q^{36} + 3 q^{37} + 6 q^{38} + 5 q^{39} - 3 q^{41} + 9 q^{42} - 7 q^{43} + 3 q^{44} - 27 q^{46} + 3 q^{47} + 5 q^{48} + 2 q^{49} - 3 q^{51} - 5 q^{52} - 6 q^{56} - q^{57} - 3 q^{58} + 3 q^{59} - 9 q^{61} + 21 q^{62} - 10 q^{63} - 2 q^{64} - 3 q^{68} - 18 q^{69} - 21 q^{71} - 6 q^{72} + 3 q^{73} - 3 q^{74} - 5 q^{75} - 8 q^{76} + 15 q^{77} - 15 q^{78} - q^{81} + 9 q^{82} + q^{84} + 21 q^{86} - 3 q^{87} + 9 q^{88} - 15 q^{89} + 5 q^{91} - 9 q^{92} + 14 q^{93} + 3 q^{94} + 9 q^{96} + q^{97} + 24 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(115\)
\(\chi(n)\) \(\zeta_{6}\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i −0.500000 + 0.866025i −1.00000 0 −1.50000 0.866025i −2.00000 + 1.73205i 1.73205i 1.00000 + 1.73205i 0
122.1 1.73205i −0.500000 0.866025i −1.00000 0 −1.50000 + 0.866025i −2.00000 1.73205i 1.73205i 1.00000 1.73205i 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
133.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 133.2.i.a 2
7.b odd 2 1 931.2.i.b 2
7.c even 3 1 931.2.p.c 2
7.c even 3 1 931.2.s.a 2
7.d odd 6 1 133.2.s.a yes 2
7.d odd 6 1 931.2.p.d 2
19.d odd 6 1 133.2.s.a yes 2
133.i even 6 1 inner 133.2.i.a 2
133.j odd 6 1 931.2.i.b 2
133.n odd 6 1 931.2.p.d 2
133.p even 6 1 931.2.s.a 2
133.s even 6 1 931.2.p.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.i.a 2 1.a even 1 1 trivial
133.2.i.a 2 133.i even 6 1 inner
133.2.s.a yes 2 7.d odd 6 1
133.2.s.a yes 2 19.d odd 6 1
931.2.i.b 2 7.b odd 2 1
931.2.i.b 2 133.j odd 6 1
931.2.p.c 2 7.c even 3 1
931.2.p.c 2 133.s even 6 1
931.2.p.d 2 7.d odd 6 1
931.2.p.d 2 133.n odd 6 1
931.2.s.a 2 7.c even 3 1
931.2.s.a 2 133.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(133, [\chi])\):

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$67$ \( T^{2} + 108 \) Copy content Toggle raw display
$71$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$79$ \( T^{2} + 12 \) Copy content Toggle raw display
$83$ \( T^{2} + 300 \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
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