Properties

Label 336.2.bl.c
Level 336
Weight 2
Character orbit 336.bl
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.bl (of order \(6\), degree \(2\), 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: yes
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 1 + \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{15} + ( 2 + 2 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} + ( -2 - \zeta_{6} ) q^{21} + ( -2 + 2 \zeta_{6} ) q^{25} + q^{27} + 9 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( -2 + \zeta_{6} ) q^{33} + ( 1 + 4 \zeta_{6} ) q^{35} -10 \zeta_{6} q^{37} + ( -6 + 12 \zeta_{6} ) q^{41} + ( 2 - 4 \zeta_{6} ) q^{43} + ( -1 - \zeta_{6} ) q^{45} -12 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( -4 + 2 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} + 3 q^{55} -2 q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + ( 3 - 2 \zeta_{6} ) q^{63} + ( -8 - 8 \zeta_{6} ) q^{67} + ( 8 - 16 \zeta_{6} ) q^{71} + ( -4 - 4 \zeta_{6} ) q^{73} -2 \zeta_{6} q^{75} + ( -4 + 5 \zeta_{6} ) q^{77} + ( 6 - 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 q^{83} + 6 q^{85} + ( -9 + 9 \zeta_{6} ) q^{87} + ( -4 + 2 \zeta_{6} ) q^{89} -5 \zeta_{6} q^{93} + ( 2 + 2 \zeta_{6} ) q^{95} + ( -11 + 22 \zeta_{6} ) q^{97} + ( 1 - 2 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 3q^{5} + q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} + 3q^{5} + q^{7} - q^{9} + 3q^{11} + 6q^{17} + 2q^{19} - 5q^{21} - 2q^{25} + 2q^{27} + 18q^{29} - 5q^{31} - 3q^{33} + 6q^{35} - 10q^{37} - 3q^{45} - 12q^{47} - 13q^{49} - 6q^{51} + 9q^{53} + 6q^{55} - 4q^{57} + 9q^{59} + 4q^{63} - 24q^{67} - 12q^{73} - 2q^{75} - 3q^{77} + 9q^{79} - q^{81} + 6q^{83} + 12q^{85} - 9q^{87} - 6q^{89} - 5q^{93} + 6q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 1.50000 0.866025i 0 0.500000 + 2.59808i 0 −0.500000 0.866025i 0
271.1 0 −0.500000 0.866025i 0 1.50000 + 0.866025i 0 0.500000 2.59808i 0 −0.500000 + 0.866025i 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
28.f 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.bl.c 2
3.b odd 2 1 1008.2.cs.e 2
4.b odd 2 1 336.2.bl.g yes 2
7.b odd 2 1 2352.2.bl.h 2
7.c even 3 1 2352.2.b.g 2
7.c even 3 1 2352.2.bl.b 2
7.d odd 6 1 336.2.bl.g yes 2
7.d odd 6 1 2352.2.b.c 2
8.b even 2 1 1344.2.bl.f 2
8.d odd 2 1 1344.2.bl.b 2
12.b even 2 1 1008.2.cs.d 2
21.g even 6 1 1008.2.cs.d 2
21.g even 6 1 7056.2.b.i 2
21.h odd 6 1 7056.2.b.e 2
28.d even 2 1 2352.2.bl.b 2
28.f even 6 1 inner 336.2.bl.c 2
28.f even 6 1 2352.2.b.g 2
28.g odd 6 1 2352.2.b.c 2
28.g odd 6 1 2352.2.bl.h 2
56.j odd 6 1 1344.2.bl.b 2
56.m even 6 1 1344.2.bl.f 2
84.j odd 6 1 1008.2.cs.e 2
84.j odd 6 1 7056.2.b.e 2
84.n even 6 1 7056.2.b.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bl.c 2 1.a even 1 1 trivial
336.2.bl.c 2 28.f even 6 1 inner
336.2.bl.g yes 2 4.b odd 2 1
336.2.bl.g yes 2 7.d odd 6 1
1008.2.cs.d 2 12.b even 2 1
1008.2.cs.d 2 21.g even 6 1
1008.2.cs.e 2 3.b odd 2 1
1008.2.cs.e 2 84.j odd 6 1
1344.2.bl.b 2 8.d odd 2 1
1344.2.bl.b 2 56.j odd 6 1
1344.2.bl.f 2 8.b even 2 1
1344.2.bl.f 2 56.m even 6 1
2352.2.b.c 2 7.d odd 6 1
2352.2.b.c 2 28.g odd 6 1
2352.2.b.g 2 7.c even 3 1
2352.2.b.g 2 28.f even 6 1
2352.2.bl.b 2 7.c even 3 1
2352.2.bl.b 2 28.d even 2 1
2352.2.bl.h 2 7.b odd 2 1
2352.2.bl.h 2 28.g odd 6 1
7056.2.b.e 2 21.h odd 6 1
7056.2.b.e 2 84.j odd 6 1
7056.2.b.i 2 21.g even 6 1
7056.2.b.i 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 \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{2} - 3 T_{5} + 3 \)
\( T_{11}^{2} - 3 T_{11} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 9 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \)
$41$ \( 1 + 26 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 74 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 61 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 24 T + 259 T^{2} + 1608 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 50 T^{2} + 5041 T^{4} \)
$73$ \( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \)
$83$ \( ( 1 - 3 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T + 101 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \)
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