Properties

Label 336.2.bc.b
Level 336
Weight 2
Character orbit 336.bc
Analytic conductor 2.683
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( -3 - 3 \zeta_{6} ) q^{11} + ( 6 - 3 \zeta_{6} ) q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + ( -2 + \zeta_{6} ) q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + ( 6 - 3 \zeta_{6} ) q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 1 + \zeta_{6} ) q^{31} + ( 9 - 9 \zeta_{6} ) q^{33} + ( -6 + 9 \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} + 6 q^{41} -4 q^{43} + 9 \zeta_{6} q^{45} + 3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -3 - 3 \zeta_{6} ) q^{51} + ( -3 - 3 \zeta_{6} ) q^{53} + ( -9 + 18 \zeta_{6} ) q^{55} -3 \zeta_{6} q^{57} + ( -3 + 3 \zeta_{6} ) q^{59} + ( -14 + 7 \zeta_{6} ) q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + ( 5 - 5 \zeta_{6} ) q^{67} + 9 \zeta_{6} q^{69} + ( 6 - 12 \zeta_{6} ) q^{71} + ( -7 - 7 \zeta_{6} ) q^{73} + ( -4 - 4 \zeta_{6} ) q^{75} + ( -3 + 15 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + 9 q^{81} + 12 q^{83} + 9 q^{85} + 9 \zeta_{6} q^{89} + ( -3 + 3 \zeta_{6} ) q^{93} + ( 3 + 3 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( 9 + 9 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{5} - 4q^{7} - 6q^{9} - 9q^{11} + 9q^{15} - 3q^{17} - 3q^{19} + 6q^{21} + 9q^{23} - 4q^{25} + 3q^{31} + 9q^{33} - 3q^{35} - 7q^{37} + 12q^{41} - 8q^{43} + 9q^{45} + 3q^{47} + 2q^{49} - 9q^{51} - 9q^{53} - 3q^{57} - 3q^{59} - 21q^{61} + 12q^{63} + 5q^{67} + 9q^{69} - 21q^{73} - 12q^{75} + 9q^{77} - q^{79} + 18q^{81} + 24q^{83} + 18q^{85} + 9q^{89} - 3q^{93} + 9q^{95} + 27q^{99} + O(q^{100}) \)

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 1.73205i 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 \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{13} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} \)
$23$ \( 1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 80 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \)
$71$ \( 1 - 34 T^{2} + 5041 T^{4} \)
$73$ \( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 146 T^{2} + 9409 T^{4} \)
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