Properties

Label 336.6.a.l
Level $336$
Weight $6$
Character orbit 336.a
Self dual yes
Analytic conductor $53.889$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 9q^{3} - 34q^{5} + 49q^{7} + 81q^{9} + O(q^{10}) \) \( q + 9q^{3} - 34q^{5} + 49q^{7} + 81q^{9} + 340q^{11} + 454q^{13} - 306q^{15} - 798q^{17} - 892q^{19} + 441q^{21} + 3192q^{23} - 1969q^{25} + 729q^{27} - 8242q^{29} + 2496q^{31} + 3060q^{33} - 1666q^{35} + 9798q^{37} + 4086q^{39} + 19834q^{41} + 17236q^{43} - 2754q^{45} - 8928q^{47} + 2401q^{49} - 7182q^{51} + 150q^{53} - 11560q^{55} - 8028q^{57} + 42396q^{59} + 14758q^{61} + 3969q^{63} - 15436q^{65} + 1676q^{67} + 28728q^{69} - 14568q^{71} + 78378q^{73} - 17721q^{75} + 16660q^{77} + 2272q^{79} + 6561q^{81} + 37764q^{83} + 27132q^{85} - 74178q^{87} - 117286q^{89} + 22246q^{91} + 22464q^{93} + 30328q^{95} + 10002q^{97} + 27540q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 9.00000 0 −34.0000 0 49.0000 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.6.a.l 1
3.b odd 2 1 1008.6.a.t 1
4.b odd 2 1 21.6.a.b 1
12.b even 2 1 63.6.a.c 1
20.d odd 2 1 525.6.a.c 1
20.e even 4 2 525.6.d.d 2
28.d even 2 1 147.6.a.e 1
28.f even 6 2 147.6.e.e 2
28.g odd 6 2 147.6.e.f 2
84.h odd 2 1 441.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 4.b odd 2 1
63.6.a.c 1 12.b even 2 1
147.6.a.e 1 28.d even 2 1
147.6.e.e 2 28.f even 6 2
147.6.e.f 2 28.g odd 6 2
336.6.a.l 1 1.a even 1 1 trivial
441.6.a.d 1 84.h odd 2 1
525.6.a.c 1 20.d odd 2 1
525.6.d.d 2 20.e even 4 2
1008.6.a.t 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} + 34 \)
\( T_{11} - 340 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -9 + T \)
$5$ \( 34 + T \)
$7$ \( -49 + T \)
$11$ \( -340 + T \)
$13$ \( -454 + T \)
$17$ \( 798 + T \)
$19$ \( 892 + T \)
$23$ \( -3192 + T \)
$29$ \( 8242 + T \)
$31$ \( -2496 + T \)
$37$ \( -9798 + T \)
$41$ \( -19834 + T \)
$43$ \( -17236 + T \)
$47$ \( 8928 + T \)
$53$ \( -150 + T \)
$59$ \( -42396 + T \)
$61$ \( -14758 + T \)
$67$ \( -1676 + T \)
$71$ \( 14568 + T \)
$73$ \( -78378 + T \)
$79$ \( -2272 + T \)
$83$ \( -37764 + T \)
$89$ \( 117286 + T \)
$97$ \( -10002 + T \)
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