Properties

Label 336.4.k.a
Level $336$
Weight $4$
Character orbit 336.k
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.k (of order \(2\), degree \(1\), 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -3 + 6 \zeta_{6} ) q^{3} + ( 19 - 18 \zeta_{6} ) q^{7} -27 q^{9} +O(q^{10})\) \( q + ( -3 + 6 \zeta_{6} ) q^{3} + ( 19 - 18 \zeta_{6} ) q^{7} -27 q^{9} + ( -36 + 72 \zeta_{6} ) q^{13} + ( -90 + 180 \zeta_{6} ) q^{19} + ( 51 + 60 \zeta_{6} ) q^{21} -125 q^{25} + ( 81 - 162 \zeta_{6} ) q^{27} + ( -90 + 180 \zeta_{6} ) q^{31} -110 q^{37} -324 q^{39} -520 q^{43} + ( 37 - 360 \zeta_{6} ) q^{49} -810 q^{57} + ( -540 + 1080 \zeta_{6} ) q^{61} + ( -513 + 486 \zeta_{6} ) q^{63} + 880 q^{67} + ( 216 - 432 \zeta_{6} ) q^{73} + ( 375 - 750 \zeta_{6} ) q^{75} -884 q^{79} + 729 q^{81} + ( 612 + 720 \zeta_{6} ) q^{91} -810 q^{93} + ( -792 + 1584 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 20q^{7} - 54q^{9} + O(q^{10}) \) \( 2q + 20q^{7} - 54q^{9} + 162q^{21} - 250q^{25} - 220q^{37} - 648q^{39} - 1040q^{43} - 286q^{49} - 1620q^{57} - 540q^{63} + 1760q^{67} - 1768q^{79} + 1458q^{81} + 1944q^{91} - 1620q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.500000 0.866025i
0.500000 + 0.866025i
0 5.19615i 0 0 0 10.0000 + 15.5885i 0 −27.0000 0
209.2 0 5.19615i 0 0 0 10.0000 15.5885i 0 −27.0000 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.k.a 2
3.b odd 2 1 CM 336.4.k.a 2
4.b odd 2 1 21.4.c.a 2
7.b odd 2 1 inner 336.4.k.a 2
12.b even 2 1 21.4.c.a 2
21.c even 2 1 inner 336.4.k.a 2
28.d even 2 1 21.4.c.a 2
28.f even 6 1 147.4.g.a 2
28.f even 6 1 147.4.g.b 2
28.g odd 6 1 147.4.g.a 2
28.g odd 6 1 147.4.g.b 2
84.h odd 2 1 21.4.c.a 2
84.j odd 6 1 147.4.g.a 2
84.j odd 6 1 147.4.g.b 2
84.n even 6 1 147.4.g.a 2
84.n even 6 1 147.4.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.c.a 2 4.b odd 2 1
21.4.c.a 2 12.b even 2 1
21.4.c.a 2 28.d even 2 1
21.4.c.a 2 84.h odd 2 1
147.4.g.a 2 28.f even 6 1
147.4.g.a 2 28.g odd 6 1
147.4.g.a 2 84.j odd 6 1
147.4.g.a 2 84.n even 6 1
147.4.g.b 2 28.f even 6 1
147.4.g.b 2 28.g odd 6 1
147.4.g.b 2 84.j odd 6 1
147.4.g.b 2 84.n even 6 1
336.4.k.a 2 1.a even 1 1 trivial
336.4.k.a 2 3.b odd 2 1 CM
336.4.k.a 2 7.b odd 2 1 inner
336.4.k.a 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 27 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 343 - 20 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 3888 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 24300 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 24300 + T^{2} \)
$37$ \( ( 110 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 520 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 874800 + T^{2} \)
$67$ \( ( -880 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 139968 + T^{2} \)
$79$ \( ( 884 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 1881792 + T^{2} \)
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