Properties

Label 336.3.o.b
Level $336$
Weight $3$
Character orbit 336.o
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(335,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.335");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.o (of order \(2\), degree \(1\), 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: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 4\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta - 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta - 1) q^{7} + 9 q^{9} + 2 \beta q^{13} - 26 q^{19} + (3 \beta + 3) q^{21} - 25 q^{25} - 27 q^{27} - 46 q^{31} - 26 q^{37} - 6 \beta q^{39} + 12 \beta q^{43} + (2 \beta - 47) q^{49} + 78 q^{57} + 14 \beta q^{61} + ( - 9 \beta - 9) q^{63} - 8 \beta q^{67} - 20 \beta q^{73} + 75 q^{75} + 10 \beta q^{79} + 81 q^{81} + ( - 2 \beta + 96) q^{91} + 138 q^{93} - 28 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 2 q^{7} + 18 q^{9} - 52 q^{19} + 6 q^{21} - 50 q^{25} - 54 q^{27} - 92 q^{31} - 52 q^{37} - 94 q^{49} + 156 q^{57} - 18 q^{63} + 150 q^{75} + 162 q^{81} + 192 q^{91} + 276 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
335.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −3.00000 0 0 0 −1.00000 6.92820i 0 9.00000 0
335.2 0 −3.00000 0 0 0 −1.00000 + 6.92820i 0 9.00000 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 CM by \(\Q(\sqrt{-3}) \)
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

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

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
\( T_{19} + 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 192 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 26)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 46)^{2} \) Copy content Toggle raw display
$37$ \( (T + 26)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 6912 \) 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
$61$ \( T^{2} + 9408 \) Copy content Toggle raw display
$67$ \( T^{2} + 3072 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 19200 \) Copy content Toggle raw display
$79$ \( T^{2} + 4800 \) 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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