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Properties

Label 336.3.be.d
Level $336$
Weight $3$
Character orbit 336.be
Analytic conductor $9.155$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,3,Mod(79,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-9,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.1364138928.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 35x^{4} + 364x^{2} + 972 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{3} + \beta_{4} q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 2) q^{7} + ( - 3 \beta_1 + 3) q^{9} + ( - \beta_{5} - \beta_{3} + \beta_{2} + \cdots + 1) q^{11} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 7) q^{13}+ \cdots + ( - 3 \beta_{4} + 3 \beta_{3} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} + q^{5} - 11 q^{7} + 9 q^{9} + 3 q^{11} - 44 q^{13} + 8 q^{17} - 30 q^{19} + 15 q^{21} + 24 q^{23} - 14 q^{25} + 34 q^{29} - 39 q^{31} - 3 q^{33} - 90 q^{35} + 6 q^{37} + 66 q^{39} - 136 q^{41}+ \cdots - 266 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 35x^{4} + 364x^{2} + 972 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 17\nu^{3} + 22\nu + 36 ) / 72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 121\nu^{3} - 36\nu^{2} - 542\nu - 396 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 23\nu^{3} + 6\nu^{2} + 118\nu + 72 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 19\nu^{2} + 2\nu + 56 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{4} - \beta_{3} - 8\beta_{2} - 4\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - \beta_{4} - 19\beta_{3} - \beta_{2} + \beta _1 + 172 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 91\beta_{4} + 17\beta_{3} + 125\beta_{2} + 151\beta _1 - 138 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
4.07390i
− 3.78298i
− 2.02297i
− 4.07390i
3.78298i
2.02297i
0 −1.50000 + 0.866025i 0 −2.29832 + 3.98081i 0 −4.95955 + 4.93992i 0 1.50000 − 2.59808i 0
79.2 0 −1.50000 + 0.866025i 0 −1.15548 + 2.00135i 0 6.36328 − 2.91696i 0 1.50000 − 2.59808i 0
79.3 0 −1.50000 + 0.866025i 0 3.95380 − 6.84819i 0 −6.90373 − 1.15694i 0 1.50000 − 2.59808i 0
319.1 0 −1.50000 − 0.866025i 0 −2.29832 − 3.98081i 0 −4.95955 − 4.93992i 0 1.50000 + 2.59808i 0
319.2 0 −1.50000 − 0.866025i 0 −1.15548 − 2.00135i 0 6.36328 + 2.91696i 0 1.50000 + 2.59808i 0
319.3 0 −1.50000 − 0.866025i 0 3.95380 + 6.84819i 0 −6.90373 + 1.15694i 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.3
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.d ✓ 6
3.b odd 2 1 1008.3.cd.h 6
4.b odd 2 1 336.3.be.f yes 6
7.c even 3 1 336.3.be.f yes 6
7.c even 3 1 2352.3.m.n 6
7.d odd 6 1 2352.3.m.o 6
12.b even 2 1 1008.3.cd.i 6
21.h odd 6 1 1008.3.cd.i 6
28.f even 6 1 2352.3.m.o 6
28.g odd 6 1 inner 336.3.be.d ✓ 6
28.g odd 6 1 2352.3.m.n 6
84.n even 6 1 1008.3.cd.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.be.d ✓ 6 1.a even 1 1 trivial
336.3.be.d ✓ 6 28.g odd 6 1 inner
336.3.be.f yes 6 4.b odd 2 1
336.3.be.f yes 6 7.c even 3 1
1008.3.cd.h 6 3.b odd 2 1
1008.3.cd.h 6 84.n even 6 1
1008.3.cd.i 6 12.b even 2 1
1008.3.cd.i 6 21.h odd 6 1
2352.3.m.n 6 7.c even 3 1
2352.3.m.n 6 28.g odd 6 1
2352.3.m.o 6 7.d odd 6 1
2352.3.m.o 6 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 \(S_{3}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{6} - T_{5}^{5} + 45T_{5}^{4} + 212T_{5}^{3} + 1852T_{5}^{2} + 3696T_{5} + 7056 \) Copy content Toggle raw display
\( T_{11}^{6} - 3T_{11}^{5} - 109T_{11}^{4} + 336T_{11}^{3} + 12580T_{11}^{2} + 4032T_{11} + 432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + \cdots + 7056 \) Copy content Toggle raw display
$7$ \( T^{6} + 11 T^{5} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + \cdots + 432 \) Copy content Toggle raw display
$13$ \( (T^{3} + 22 T^{2} + \cdots - 4312)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 8 T^{5} + \cdots + 20358144 \) Copy content Toggle raw display
$19$ \( T^{6} + 30 T^{5} + \cdots + 1138368 \) Copy content Toggle raw display
$23$ \( T^{6} - 24 T^{5} + \cdots + 66382848 \) Copy content Toggle raw display
$29$ \( (T^{3} - 17 T^{2} + \cdots + 35952)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 39 T^{5} + \cdots + 3756483 \) Copy content Toggle raw display
$37$ \( T^{6} - 6 T^{5} + \cdots + 183115024 \) Copy content Toggle raw display
$41$ \( (T^{3} + 68 T^{2} + \cdots - 37152)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 4010 T^{4} + \cdots + 7660812 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 7468832448 \) Copy content Toggle raw display
$53$ \( T^{6} + 89 T^{5} + \cdots + 609892416 \) Copy content Toggle raw display
$59$ \( T^{6} - 63 T^{5} + \cdots + 84672 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 73796982336 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 84665952108 \) Copy content Toggle raw display
$71$ \( T^{6} + 6164 T^{4} + \cdots + 615874752 \) Copy content Toggle raw display
$73$ \( T^{6} - 36 T^{5} + \cdots + 688642564 \) Copy content Toggle raw display
$79$ \( T^{6} - 3 T^{5} + \cdots + 140726403 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 22314082608 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 10788229287936 \) Copy content Toggle raw display
$97$ \( (T^{3} + 133 T^{2} + \cdots + 35252)^{2} \) Copy content Toggle raw display
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