Properties

Label 156.2.u.a
Level $156$
Weight $2$
Character orbit 156.u
Analytic conductor $1.246$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,2,Mod(41,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.u (of order \(12\), 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: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9} + (4 \zeta_{12}^{3} - \zeta_{12}) q^{13} + ( - 2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{19} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{21} - 5 \zeta_{12}^{3} q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + ( - 5 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 5) q^{31} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 7 \zeta_{12} + 4) q^{37} + (2 \zeta_{12}^{2} - 7) q^{39} + ( - 7 \zeta_{12}^{2} - 7) q^{43} + ( - 7 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 7 \zeta_{12} - 10) q^{49} + ( - 7 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 7) q^{57} + (18 \zeta_{12}^{3} - 9 \zeta_{12}) q^{61} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 9 \zeta_{12} + 3) q^{63} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 7 \zeta_{12} + 9) q^{67} + ( - \zeta_{12}^{3} + 9 \zeta_{12}^{2} + 9 \zeta_{12} - 1) q^{73} + ( - 5 \zeta_{12}^{2} + 10) q^{75} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{79} - 9 \zeta_{12}^{2} q^{81} + ( - \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 10 \zeta_{12} - 11) q^{91} + (7 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 4 \zeta_{12} + 11) q^{93} + ( - 3 \zeta_{12}^{3} - 11 \zeta_{12}^{2} - 8 \zeta_{12} + 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{7} - 6 q^{9} + 2 q^{19} - 6 q^{21} + 22 q^{31} + 22 q^{37} - 24 q^{39} - 42 q^{43} - 30 q^{49} + 12 q^{57} + 24 q^{63} + 32 q^{67} + 14 q^{73} + 30 q^{75} - 18 q^{81} - 34 q^{91} + 30 q^{93} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0.866025 + 1.50000i 0 0 0 2.23205 0.598076i 0 −1.50000 + 2.59808i 0
89.1 0 −0.866025 1.50000i 0 0 0 −1.23205 4.59808i 0 −1.50000 + 2.59808i 0
137.1 0 0.866025 1.50000i 0 0 0 2.23205 + 0.598076i 0 −1.50000 2.59808i 0
149.1 0 −0.866025 + 1.50000i 0 0 0 −1.23205 + 4.59808i 0 −1.50000 2.59808i 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 CM by \(\Q(\sqrt{-3}) \)
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.u.a 4
3.b odd 2 1 CM 156.2.u.a 4
4.b odd 2 1 624.2.cn.a 4
12.b even 2 1 624.2.cn.a 4
13.f odd 12 1 inner 156.2.u.a 4
39.k even 12 1 inner 156.2.u.a 4
52.l even 12 1 624.2.cn.a 4
156.v odd 12 1 624.2.cn.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.u.a 4 1.a even 1 1 trivial
156.2.u.a 4 3.b odd 2 1 CM
156.2.u.a 4 13.f odd 12 1 inner
156.2.u.a 4 39.k even 12 1 inner
624.2.cn.a 4 4.b odd 2 1
624.2.cn.a 4 12.b even 2 1
624.2.cn.a 4 52.l even 12 1
624.2.cn.a 4 156.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 17 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + 50 T^{2} - 364 T + 676 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$37$ \( T^{4} - 22 T^{3} + 122 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 243 T^{2} + 59049 \) Copy content Toggle raw display
$67$ \( T^{4} - 32 T^{3} + 377 T^{2} + \cdots + 11881 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + 98 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
$79$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 10 T^{3} + 221 T^{2} + \cdots + 27889 \) Copy content Toggle raw display
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