Properties

Label 336.3.z.c
Level $336$
Weight $3$
Character orbit 336.z
Analytic conductor $9.155$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,3,Mod(47,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.z (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: \(9.15533688251\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{3} + (5 \zeta_{6} - 8) q^{7} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{3} + (5 \zeta_{6} - 8) q^{7} - 9 \zeta_{6} q^{9} + ( - 30 \zeta_{6} + 15) q^{13} - 37 \zeta_{6} q^{19} + (24 \zeta_{6} - 9) q^{21} + ( - 25 \zeta_{6} + 25) q^{25} - 27 q^{27} + (59 \zeta_{6} - 59) q^{31} + 47 \zeta_{6} q^{37} + ( - 45 \zeta_{6} - 45) q^{39} + ( - 26 \zeta_{6} + 13) q^{43} + ( - 55 \zeta_{6} + 39) q^{49} - 111 q^{57} + ( - 56 \zeta_{6} + 112) q^{61} + (27 \zeta_{6} + 45) q^{63} + (45 \zeta_{6} + 45) q^{67} + ( - 17 \zeta_{6} - 17) q^{73} - 75 \zeta_{6} q^{75} + ( - 51 \zeta_{6} + 102) q^{79} + (81 \zeta_{6} - 81) q^{81} + (165 \zeta_{6} + 30) q^{91} + 177 \zeta_{6} q^{93} + ( - 224 \zeta_{6} + 112) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 11 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 11 q^{7} - 9 q^{9} - 37 q^{19} + 6 q^{21} + 25 q^{25} - 54 q^{27} - 59 q^{31} + 47 q^{37} - 135 q^{39} + 23 q^{49} - 222 q^{57} + 168 q^{61} + 117 q^{63} + 135 q^{67} - 51 q^{73} - 75 q^{75} + 153 q^{79} - 81 q^{81} + 225 q^{91} + 177 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.50000 + 2.59808i 0 0 0 −5.50000 4.33013i 0 −4.50000 + 7.79423i 0
143.1 0 1.50000 2.59808i 0 0 0 −5.50000 + 4.33013i 0 −4.50000 7.79423i 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 \(\Q(\sqrt{-3}) \)
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.z.c yes 2
3.b odd 2 1 CM 336.3.z.c yes 2
4.b odd 2 1 336.3.z.b 2
7.d odd 6 1 336.3.z.b 2
12.b even 2 1 336.3.z.b 2
21.g even 6 1 336.3.z.b 2
28.f even 6 1 inner 336.3.z.c yes 2
84.j odd 6 1 inner 336.3.z.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.z.b 2 4.b odd 2 1
336.3.z.b 2 7.d odd 6 1
336.3.z.b 2 12.b even 2 1
336.3.z.b 2 21.g even 6 1
336.3.z.c yes 2 1.a even 1 1 trivial
336.3.z.c yes 2 3.b odd 2 1 CM
336.3.z.c yes 2 28.f even 6 1 inner
336.3.z.c yes 2 84.j odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13}^{2} + 675 \) Copy content Toggle raw display
\( T_{19}^{2} + 37T_{19} + 1369 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 11T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 675 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 37T + 1369 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$37$ \( T^{2} - 47T + 2209 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 507 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 168T + 9408 \) Copy content Toggle raw display
$67$ \( T^{2} - 135T + 6075 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 51T + 867 \) Copy content Toggle raw display
$79$ \( T^{2} - 153T + 7803 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 37632 \) Copy content Toggle raw display
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