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Properties

Label 336.3.bn.e
Level $336$
Weight $3$
Character orbit 336.bn
Analytic conductor $9.155$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,3,Mod(65,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.65"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content 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: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{3} + 3 \beta_1 q^{5} + (5 \beta_{2} - 8) q^{7} + ( - \beta_{2} + 4 \beta_1 + 1) q^{9} + (3 \beta_{3} - 3 \beta_1) q^{11} + 14 q^{13} + (6 \beta_{3} + 15) q^{15}+ \cdots + (3 \beta_{3} - 60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 22 q^{7} + 2 q^{9} + 56 q^{13} + 60 q^{15} + 16 q^{19} - 52 q^{21} + 40 q^{25} + 88 q^{27} - 62 q^{31} - 30 q^{33} + 56 q^{37} + 56 q^{39} + 208 q^{43} + 120 q^{45} + 46 q^{49} - 120 q^{51}+ \cdots - 240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.93649 − 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 − 1.11803i
0 −0.936492 + 2.85008i 0 −5.80948 − 3.35410i 0 −5.50000 + 4.33013i 0 −7.24597 − 5.33816i 0
65.2 0 2.93649 + 0.614017i 0 5.80948 + 3.35410i 0 −5.50000 + 4.33013i 0 8.24597 + 3.60611i 0
305.1 0 −0.936492 − 2.85008i 0 −5.80948 + 3.35410i 0 −5.50000 − 4.33013i 0 −7.24597 + 5.33816i 0
305.2 0 2.93649 − 0.614017i 0 5.80948 − 3.35410i 0 −5.50000 − 4.33013i 0 8.24597 − 3.60611i 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 inner
7.c even 3 1 inner
21.h 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.bn.e 4
3.b odd 2 1 inner 336.3.bn.e 4
4.b odd 2 1 84.3.p.b ✓ 4
7.c even 3 1 inner 336.3.bn.e 4
12.b even 2 1 84.3.p.b ✓ 4
21.h odd 6 1 inner 336.3.bn.e 4
28.d even 2 1 588.3.p.e 4
28.f even 6 1 588.3.c.d 2
28.f even 6 1 588.3.p.e 4
28.g odd 6 1 84.3.p.b ✓ 4
28.g odd 6 1 588.3.c.g 2
84.h odd 2 1 588.3.p.e 4
84.j odd 6 1 588.3.c.d 2
84.j odd 6 1 588.3.p.e 4
84.n even 6 1 84.3.p.b ✓ 4
84.n even 6 1 588.3.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.p.b ✓ 4 4.b odd 2 1
84.3.p.b ✓ 4 12.b even 2 1
84.3.p.b ✓ 4 28.g odd 6 1
84.3.p.b ✓ 4 84.n even 6 1
336.3.bn.e 4 1.a even 1 1 trivial
336.3.bn.e 4 3.b odd 2 1 inner
336.3.bn.e 4 7.c even 3 1 inner
336.3.bn.e 4 21.h odd 6 1 inner
588.3.c.d 2 28.f even 6 1
588.3.c.d 2 84.j odd 6 1
588.3.c.g 2 28.g odd 6 1
588.3.c.g 2 84.n even 6 1
588.3.p.e 4 28.d even 2 1
588.3.p.e 4 28.f even 6 1
588.3.p.e 4 84.h odd 2 1
588.3.p.e 4 84.j odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 45T_{5}^{2} + 2025 \) Copy content Toggle raw display
\( T_{13} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{4} - 45T^{2} + 2025 \) Copy content Toggle raw display
$7$ \( (T^{2} + 11 T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 45T^{2} + 2025 \) Copy content Toggle raw display
$13$ \( (T - 14)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 720 T^{2} + 518400 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 180 T^{2} + 32400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2205)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 31 T + 961)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 28 T + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4500)^{2} \) Copy content Toggle raw display
$43$ \( (T - 52)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 1620 T^{2} + 2624400 \) Copy content Toggle raw display
$53$ \( T^{4} - 45T^{2} + 2025 \) Copy content Toggle raw display
$59$ \( T^{4} - 405 T^{2} + 164025 \) Copy content Toggle raw display
$61$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1620)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 98 T + 9604)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 101 T + 10201)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 7605)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 4500 T^{2} + 20250000 \) Copy content Toggle raw display
$97$ \( (T + 13)^{4} \) Copy content Toggle raw display
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