Properties

Label 336.2.bc.c
Level 336
Weight 2
Character orbit 336.bc
Analytic conductor 2.683
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{6} ) q^{3} + ( -2 + 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{6} ) q^{3} + ( -2 + 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -1 + 2 \zeta_{6} ) q^{13} + ( 6 - 3 \zeta_{6} ) q^{19} + ( -5 + 4 \zeta_{6} ) q^{21} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -5 - 5 \zeta_{6} ) q^{31} -\zeta_{6} q^{37} + ( -3 + 3 \zeta_{6} ) q^{39} + 5 q^{43} + ( -5 - 3 \zeta_{6} ) q^{49} + 9 q^{57} + ( 8 - 4 \zeta_{6} ) q^{61} + ( -9 + 3 \zeta_{6} ) q^{63} + ( 11 - 11 \zeta_{6} ) q^{67} + ( -9 - 9 \zeta_{6} ) q^{73} + ( 10 - 5 \zeta_{6} ) q^{75} -13 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -4 - \zeta_{6} ) q^{91} -15 \zeta_{6} q^{93} + ( -8 + 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} - q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} - q^{7} + 3q^{9} + 9q^{19} - 6q^{21} + 5q^{25} - 15q^{31} - q^{37} - 3q^{39} + 10q^{43} - 13q^{49} + 18q^{57} + 12q^{61} - 15q^{63} + 11q^{67} - 27q^{73} + 15q^{75} - 13q^{79} - 9q^{81} - 9q^{91} - 15q^{93} + O(q^{100}) \)

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 1.50000 + 0.866025i 0 0 0 −0.500000 + 2.59808i 0 1.50000 + 2.59808i 0
257.1 0 1.50000 0.866025i 0 0 0 −0.500000 2.59808i 0 1.50000 2.59808i 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 336.2.bc.c 2
3.b odd 2 1 CM 336.2.bc.c 2
4.b odd 2 1 21.2.g.a 2
7.c even 3 1 2352.2.k.c 2
7.d odd 6 1 inner 336.2.bc.c 2
7.d odd 6 1 2352.2.k.c 2
12.b even 2 1 21.2.g.a 2
20.d odd 2 1 525.2.t.c 2
20.e even 4 2 525.2.q.d 4
21.g even 6 1 inner 336.2.bc.c 2
21.g even 6 1 2352.2.k.c 2
21.h odd 6 1 2352.2.k.c 2
28.d even 2 1 147.2.g.a 2
28.f even 6 1 21.2.g.a 2
28.f even 6 1 147.2.c.a 2
28.g odd 6 1 147.2.c.a 2
28.g odd 6 1 147.2.g.a 2
36.f odd 6 1 567.2.i.b 2
36.f odd 6 1 567.2.s.a 2
36.h even 6 1 567.2.i.b 2
36.h even 6 1 567.2.s.a 2
60.h even 2 1 525.2.t.c 2
60.l odd 4 2 525.2.q.d 4
84.h odd 2 1 147.2.g.a 2
84.j odd 6 1 21.2.g.a 2
84.j odd 6 1 147.2.c.a 2
84.n even 6 1 147.2.c.a 2
84.n even 6 1 147.2.g.a 2
140.s even 6 1 525.2.t.c 2
140.x odd 12 2 525.2.q.d 4
252.n even 6 1 567.2.i.b 2
252.r odd 6 1 567.2.s.a 2
252.bj even 6 1 567.2.s.a 2
252.bn odd 6 1 567.2.i.b 2
420.be odd 6 1 525.2.t.c 2
420.br even 12 2 525.2.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.g.a 2 4.b odd 2 1
21.2.g.a 2 12.b even 2 1
21.2.g.a 2 28.f even 6 1
21.2.g.a 2 84.j odd 6 1
147.2.c.a 2 28.f even 6 1
147.2.c.a 2 28.g odd 6 1
147.2.c.a 2 84.j odd 6 1
147.2.c.a 2 84.n even 6 1
147.2.g.a 2 28.d even 2 1
147.2.g.a 2 28.g odd 6 1
147.2.g.a 2 84.h odd 2 1
147.2.g.a 2 84.n even 6 1
336.2.bc.c 2 1.a even 1 1 trivial
336.2.bc.c 2 3.b odd 2 1 CM
336.2.bc.c 2 7.d odd 6 1 inner
336.2.bc.c 2 21.g even 6 1 inner
525.2.q.d 4 20.e even 4 2
525.2.q.d 4 60.l odd 4 2
525.2.q.d 4 140.x odd 12 2
525.2.q.d 4 420.br even 12 2
525.2.t.c 2 20.d odd 2 1
525.2.t.c 2 60.h even 2 1
525.2.t.c 2 140.s even 6 1
525.2.t.c 2 420.be odd 6 1
567.2.i.b 2 36.f odd 6 1
567.2.i.b 2 36.h even 6 1
567.2.i.b 2 252.n even 6 1
567.2.i.b 2 252.bn odd 6 1
567.2.s.a 2 36.f odd 6 1
567.2.s.a 2 36.h even 6 1
567.2.s.a 2 252.r odd 6 1
567.2.s.a 2 252.bj even 6 1
2352.2.k.c 2 7.c even 3 1
2352.2.k.c 2 7.d odd 6 1
2352.2.k.c 2 21.g even 6 1
2352.2.k.c 2 21.h odd 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{13}^{2} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \)
$23$ \( 1 + 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 53 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 + 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
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