Properties

Label 336.4.k.a
Level $336$
Weight $4$
Character orbit 336.k
Analytic conductor $19.825$
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] = [336,4,Mod(209,336)]
 
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([0, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.209");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.k (of order \(2\), degree \(1\), not 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: \(19.8246417619\)
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: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 21)
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 \(\beta = 3\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( - 3 \beta + 10) q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + ( - 3 \beta + 10) q^{7} - 27 q^{9} + 12 \beta q^{13} + 30 \beta q^{19} + (10 \beta + 81) q^{21} - 125 q^{25} - 27 \beta q^{27} + 30 \beta q^{31} - 110 q^{37} - 324 q^{39} - 520 q^{43} + ( - 60 \beta - 143) q^{49} - 810 q^{57} + 180 \beta q^{61} + (81 \beta - 270) q^{63} + 880 q^{67} - 72 \beta q^{73} - 125 \beta q^{75} - 884 q^{79} + 729 q^{81} + (120 \beta + 972) q^{91} - 810 q^{93} + 264 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{7} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{7} - 54 q^{9} + 162 q^{21} - 250 q^{25} - 220 q^{37} - 648 q^{39} - 1040 q^{43} - 286 q^{49} - 1620 q^{57} - 540 q^{63} + 1760 q^{67} - 1768 q^{79} + 1458 q^{81} + 1944 q^{91} - 1620 q^{93}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.500000 0.866025i
0.500000 + 0.866025i
0 5.19615i 0 0 0 10.0000 + 15.5885i 0 −27.0000 0
209.2 0 5.19615i 0 0 0 10.0000 15.5885i 0 −27.0000 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 \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.k.a 2
3.b odd 2 1 CM 336.4.k.a 2
4.b odd 2 1 21.4.c.a 2
7.b odd 2 1 inner 336.4.k.a 2
12.b even 2 1 21.4.c.a 2
21.c even 2 1 inner 336.4.k.a 2
28.d even 2 1 21.4.c.a 2
28.f even 6 1 147.4.g.a 2
28.f even 6 1 147.4.g.b 2
28.g odd 6 1 147.4.g.a 2
28.g odd 6 1 147.4.g.b 2
84.h odd 2 1 21.4.c.a 2
84.j odd 6 1 147.4.g.a 2
84.j odd 6 1 147.4.g.b 2
84.n even 6 1 147.4.g.a 2
84.n even 6 1 147.4.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.c.a 2 4.b odd 2 1
21.4.c.a 2 12.b even 2 1
21.4.c.a 2 28.d even 2 1
21.4.c.a 2 84.h odd 2 1
147.4.g.a 2 28.f even 6 1
147.4.g.a 2 28.g odd 6 1
147.4.g.a 2 84.j odd 6 1
147.4.g.a 2 84.n even 6 1
147.4.g.b 2 28.f even 6 1
147.4.g.b 2 28.g odd 6 1
147.4.g.b 2 84.j odd 6 1
147.4.g.b 2 84.n even 6 1
336.4.k.a 2 1.a even 1 1 trivial
336.4.k.a 2 3.b odd 2 1 CM
336.4.k.a 2 7.b odd 2 1 inner
336.4.k.a 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 27 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 20T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3888 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 24300 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 24300 \) Copy content Toggle raw display
$37$ \( (T + 110)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 520)^{2} \) 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
$61$ \( T^{2} + 874800 \) Copy content Toggle raw display
$67$ \( (T - 880)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 139968 \) Copy content Toggle raw display
$79$ \( (T + 884)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1881792 \) Copy content Toggle raw display
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