Properties

Label 336.4.bc.b
Level $336$
Weight $4$
Character orbit 336.bc
Analytic conductor $19.825$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 3 + 3 \zeta_{6} ) q^{3} + ( 18 - 19 \zeta_{6} ) q^{7} + 27 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 3 + 3 \zeta_{6} ) q^{3} + ( 18 - 19 \zeta_{6} ) q^{7} + 27 \zeta_{6} q^{9} + ( -53 + 106 \zeta_{6} ) q^{13} + ( 146 - 73 \zeta_{6} ) q^{19} + ( 111 - 60 \zeta_{6} ) q^{21} + ( 125 - 125 \zeta_{6} ) q^{25} + ( -81 + 162 \zeta_{6} ) q^{27} + ( 109 + 109 \zeta_{6} ) q^{31} + 323 \zeta_{6} q^{37} + ( -477 + 477 \zeta_{6} ) q^{39} + 71 q^{43} + ( -37 - 323 \zeta_{6} ) q^{49} + 657 q^{57} + ( -1080 + 540 \zeta_{6} ) q^{61} + ( 513 - 27 \zeta_{6} ) q^{63} + ( -127 + 127 \zeta_{6} ) q^{67} + ( 703 + 703 \zeta_{6} ) q^{73} + ( 750 - 375 \zeta_{6} ) q^{75} -1387 \zeta_{6} q^{79} + ( -729 + 729 \zeta_{6} ) q^{81} + ( 1060 + 901 \zeta_{6} ) q^{91} + 981 \zeta_{6} q^{93} + ( 792 - 1584 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 9q^{3} + 17q^{7} + 27q^{9} + O(q^{10}) \) \( 2q + 9q^{3} + 17q^{7} + 27q^{9} + 219q^{19} + 162q^{21} + 125q^{25} + 327q^{31} + 323q^{37} - 477q^{39} + 142q^{43} - 397q^{49} + 1314q^{57} - 1620q^{61} + 999q^{63} - 127q^{67} + 2109q^{73} + 1125q^{75} - 1387q^{79} - 729q^{81} + 3021q^{91} + 981q^{93} + O(q^{100}) \)

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.

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 4.50000 + 2.59808i 0 0 0 8.50000 16.4545i 0 13.5000 + 23.3827i 0
257.1 0 4.50000 2.59808i 0 0 0 8.50000 + 16.4545i 0 13.5000 23.3827i 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.d odd 6 1 inner
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 336.4.bc.b 2
3.b odd 2 1 CM 336.4.bc.b 2
4.b odd 2 1 84.4.k.a 2
7.d odd 6 1 inner 336.4.bc.b 2
12.b even 2 1 84.4.k.a 2
21.g even 6 1 inner 336.4.bc.b 2
28.d even 2 1 588.4.k.b 2
28.f even 6 1 84.4.k.a 2
28.f even 6 1 588.4.f.a 2
28.g odd 6 1 588.4.f.a 2
28.g odd 6 1 588.4.k.b 2
84.h odd 2 1 588.4.k.b 2
84.j odd 6 1 84.4.k.a 2
84.j odd 6 1 588.4.f.a 2
84.n even 6 1 588.4.f.a 2
84.n even 6 1 588.4.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.k.a 2 4.b odd 2 1
84.4.k.a 2 12.b even 2 1
84.4.k.a 2 28.f even 6 1
84.4.k.a 2 84.j odd 6 1
336.4.bc.b 2 1.a even 1 1 trivial
336.4.bc.b 2 3.b odd 2 1 CM
336.4.bc.b 2 7.d odd 6 1 inner
336.4.bc.b 2 21.g even 6 1 inner
588.4.f.a 2 28.f even 6 1
588.4.f.a 2 28.g odd 6 1
588.4.f.a 2 84.j odd 6 1
588.4.f.a 2 84.n even 6 1
588.4.k.b 2 28.d even 2 1
588.4.k.b 2 28.g odd 6 1
588.4.k.b 2 84.h odd 2 1
588.4.k.b 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):

\( T_{5} \)
\( T_{13}^{2} + 8427 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 27 - 9 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 343 - 17 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 8427 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 15987 - 219 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 35643 - 327 T + T^{2} \)
$37$ \( 104329 - 323 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -71 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 874800 + 1620 T + T^{2} \)
$67$ \( 16129 + 127 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1482627 - 2109 T + T^{2} \)
$79$ \( 1923769 + 1387 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 1881792 + T^{2} \)
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