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Properties

Label 336.3.be.b
Level $336$
Weight $3$
Character orbit 336.be
Analytic conductor $9.155$
Analytic rank $0$
Dimension $4$
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] = [336,3,Mod(79,336)]
 
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, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.be (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 6) q^{7} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 6) q^{7} + 3 \beta_{2} q^{9} + (2 \beta_{2} - 5 \beta_1 + 7) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 + 7) q^{13} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{15} + ( - 8 \beta_{3} - 12 \beta_{2} + \cdots + 8) q^{17}+ \cdots + (15 \beta_{3} + 27 \beta_{2} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + q^{5} - 12 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + q^{5} - 12 q^{7} + 6 q^{9} + 27 q^{11} + 26 q^{13} + 20 q^{17} + 9 q^{19} - 27 q^{21} - 12 q^{23} + 21 q^{25} - 2 q^{29} + 90 q^{31} + 27 q^{33} - 36 q^{35} - 19 q^{37} + 39 q^{39} - 4 q^{41} - 3 q^{45} - 84 q^{47} - 10 q^{49} + 60 q^{51} + 169 q^{53} + 18 q^{57} - 27 q^{59} + 96 q^{61} - 45 q^{63} - 22 q^{65} - 9 q^{67} - 24 q^{69} - 191 q^{73} + 63 q^{75} - 169 q^{77} + 168 q^{79} - 18 q^{81} - 208 q^{85} - 3 q^{87} + 122 q^{89} - 135 q^{91} + 90 q^{93} - 366 q^{95} + 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu + 5 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} + 4\beta _1 + 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−1.63746 + 1.52274i
2.13746 − 0.656712i
−1.63746 − 1.52274i
2.13746 + 0.656712i
0 1.50000 − 0.866025i 0 −1.63746 + 2.83616i 0 −6.77492 + 1.76082i 0 1.50000 − 2.59808i 0
79.2 0 1.50000 − 0.866025i 0 2.13746 − 3.70219i 0 0.774917 − 6.95698i 0 1.50000 − 2.59808i 0
319.1 0 1.50000 + 0.866025i 0 −1.63746 − 2.83616i 0 −6.77492 − 1.76082i 0 1.50000 + 2.59808i 0
319.2 0 1.50000 + 0.866025i 0 2.13746 + 3.70219i 0 0.774917 + 6.95698i 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
28.g 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.be.b yes 4
3.b odd 2 1 1008.3.cd.f 4
4.b odd 2 1 336.3.be.a ✓ 4
7.c even 3 1 336.3.be.a ✓ 4
7.c even 3 1 2352.3.m.g 4
7.d odd 6 1 2352.3.m.h 4
12.b even 2 1 1008.3.cd.g 4
21.h odd 6 1 1008.3.cd.g 4
28.f even 6 1 2352.3.m.h 4
28.g odd 6 1 inner 336.3.be.b yes 4
28.g odd 6 1 2352.3.m.g 4
84.n even 6 1 1008.3.cd.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.be.a ✓ 4 4.b odd 2 1
336.3.be.a ✓ 4 7.c even 3 1
336.3.be.b yes 4 1.a even 1 1 trivial
336.3.be.b yes 4 28.g odd 6 1 inner
1008.3.cd.f 4 3.b odd 2 1
1008.3.cd.f 4 84.n even 6 1
1008.3.cd.g 4 12.b even 2 1
1008.3.cd.g 4 21.h odd 6 1
2352.3.m.g 4 7.c even 3 1
2352.3.m.g 4 28.g odd 6 1
2352.3.m.h 4 7.d odd 6 1
2352.3.m.h 4 28.f even 6 1

Hecke kernels

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

\( T_{5}^{4} - T_{5}^{3} + 15T_{5}^{2} + 14T_{5} + 196 \) Copy content Toggle raw display
\( T_{11}^{4} - 27T_{11}^{3} + 185T_{11}^{2} + 1566T_{11} + 3364 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{4} + 12 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{4} - 27 T^{3} + \cdots + 3364 \) Copy content Toggle raw display
$13$ \( (T^{2} - 13 T + 28)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 20 T^{3} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( T^{4} - 9 T^{3} + \cdots + 633616 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{2} + T - 1724)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 90 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$37$ \( T^{4} + 19 T^{3} + \cdots + 369664 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T - 1424)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 3839 T^{2} + 2829124 \) Copy content Toggle raw display
$47$ \( (T^{2} + 42 T + 588)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 169 T^{3} + \cdots + 46022656 \) Copy content Toggle raw display
$59$ \( T^{4} + 27 T^{3} + \cdots + 1721344 \) Copy content Toggle raw display
$61$ \( T^{4} - 96 T^{3} + \cdots + 4309776 \) Copy content Toggle raw display
$67$ \( T^{4} + 9 T^{3} + \cdots + 142884 \) Copy content Toggle raw display
$71$ \( T^{4} + 15416 T^{2} + 56130064 \) Copy content Toggle raw display
$73$ \( T^{4} + 191 T^{3} + \cdots + 82919236 \) Copy content Toggle raw display
$79$ \( T^{4} - 168 T^{3} + \cdots + 660969 \) Copy content Toggle raw display
$83$ \( T^{4} + 25559 T^{2} + 60124516 \) Copy content Toggle raw display
$89$ \( T^{4} - 122 T^{3} + \cdots + 13424896 \) Copy content Toggle raw display
$97$ \( (T^{2} - 25 T - 13538)^{2} \) Copy content Toggle raw display
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