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Properties

Label 336.2.bc.b
Level $336$
Weight $2$
Character orbit 336.bc
Analytic conductor $2.683$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,2,Mod(17,336)] 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([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-3] 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: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 1) q^{3} - 3 \zeta_{6} q^{5} + ( - 2 \zeta_{6} - 1) q^{7} - 3 q^{9} + ( - 3 \zeta_{6} - 3) q^{11} + ( - 3 \zeta_{6} + 6) q^{15} + (3 \zeta_{6} - 3) q^{17} + (\zeta_{6} - 2) q^{19} + ( - 4 \zeta_{6} + 5) q^{21} + \cdots + (9 \zeta_{6} + 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - 4 q^{7} - 6 q^{9} - 9 q^{11} + 9 q^{15} - 3 q^{17} - 3 q^{19} + 6 q^{21} + 9 q^{23} - 4 q^{25} + 3 q^{31} + 9 q^{33} - 3 q^{35} - 7 q^{37} + 12 q^{41} - 8 q^{43} + 9 q^{45} + 3 q^{47} + 2 q^{49}+ \cdots + 27 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 1.73205i 0 −1.50000 − 2.59808i 0 −2.00000 − 1.73205i 0 −3.00000 0
257.1 0 − 1.73205i 0 −1.50000 + 2.59808i 0 −2.00000 + 1.73205i 0 −3.00000 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
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bc.b 2
3.b odd 2 1 336.2.bc.d 2
4.b odd 2 1 84.2.k.b yes 2
7.c even 3 1 2352.2.k.d 2
7.d odd 6 1 336.2.bc.d 2
7.d odd 6 1 2352.2.k.a 2
12.b even 2 1 84.2.k.a ✓ 2
20.d odd 2 1 2100.2.bi.e 2
20.e even 4 2 2100.2.bo.f 4
21.g even 6 1 inner 336.2.bc.b 2
21.g even 6 1 2352.2.k.d 2
21.h odd 6 1 2352.2.k.a 2
28.d even 2 1 588.2.k.c 2
28.f even 6 1 84.2.k.a ✓ 2
28.f even 6 1 588.2.f.c 2
28.g odd 6 1 588.2.f.a 2
28.g odd 6 1 588.2.k.d 2
36.f odd 6 1 2268.2.w.a 2
36.f odd 6 1 2268.2.bm.f 2
36.h even 6 1 2268.2.w.f 2
36.h even 6 1 2268.2.bm.a 2
60.h even 2 1 2100.2.bi.f 2
60.l odd 4 2 2100.2.bo.a 4
84.h odd 2 1 588.2.k.d 2
84.j odd 6 1 84.2.k.b yes 2
84.j odd 6 1 588.2.f.a 2
84.n even 6 1 588.2.f.c 2
84.n even 6 1 588.2.k.c 2
140.s even 6 1 2100.2.bi.f 2
140.x odd 12 2 2100.2.bo.a 4
252.n even 6 1 2268.2.w.f 2
252.r odd 6 1 2268.2.bm.f 2
252.bj even 6 1 2268.2.bm.a 2
252.bn odd 6 1 2268.2.w.a 2
420.be odd 6 1 2100.2.bi.e 2
420.br even 12 2 2100.2.bo.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.a ✓ 2 12.b even 2 1
84.2.k.a ✓ 2 28.f even 6 1
84.2.k.b yes 2 4.b odd 2 1
84.2.k.b yes 2 84.j odd 6 1
336.2.bc.b 2 1.a even 1 1 trivial
336.2.bc.b 2 21.g even 6 1 inner
336.2.bc.d 2 3.b odd 2 1
336.2.bc.d 2 7.d odd 6 1
588.2.f.a 2 28.g odd 6 1
588.2.f.a 2 84.j odd 6 1
588.2.f.c 2 28.f even 6 1
588.2.f.c 2 84.n even 6 1
588.2.k.c 2 28.d even 2 1
588.2.k.c 2 84.n even 6 1
588.2.k.d 2 28.g odd 6 1
588.2.k.d 2 84.h odd 2 1
2100.2.bi.e 2 20.d odd 2 1
2100.2.bi.e 2 420.be odd 6 1
2100.2.bi.f 2 60.h even 2 1
2100.2.bi.f 2 140.s even 6 1
2100.2.bo.a 4 60.l odd 4 2
2100.2.bo.a 4 140.x odd 12 2
2100.2.bo.f 4 20.e even 4 2
2100.2.bo.f 4 420.br even 12 2
2268.2.w.a 2 36.f odd 6 1
2268.2.w.a 2 252.bn odd 6 1
2268.2.w.f 2 36.h even 6 1
2268.2.w.f 2 252.n even 6 1
2268.2.bm.a 2 36.h even 6 1
2268.2.bm.a 2 252.bj even 6 1
2268.2.bm.f 2 36.f odd 6 1
2268.2.bm.f 2 252.r odd 6 1
2352.2.k.a 2 7.d odd 6 1
2352.2.k.a 2 21.h odd 6 1
2352.2.k.d 2 7.c even 3 1
2352.2.k.d 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$37$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( T^{2} + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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