Properties

Label 336.6.a.i
Level $336$
Weight $6$
Character orbit 336.a
Self dual yes
Analytic conductor $53.889$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 9 q^{3} + 94 q^{5} + 49 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + 94 q^{5} + 49 q^{7} + 81 q^{9} - 52 q^{11} - 770 q^{13} - 846 q^{15} - 2022 q^{17} - 1732 q^{19} - 441 q^{21} + 576 q^{23} + 5711 q^{25} - 729 q^{27} + 5518 q^{29} - 6336 q^{31} + 468 q^{33} + 4606 q^{35} - 7338 q^{37} + 6930 q^{39} - 3262 q^{41} - 5420 q^{43} + 7614 q^{45} - 864 q^{47} + 2401 q^{49} + 18198 q^{51} + 4182 q^{53} - 4888 q^{55} + 15588 q^{57} + 11220 q^{59} - 45602 q^{61} + 3969 q^{63} - 72380 q^{65} - 1396 q^{67} - 5184 q^{69} - 18720 q^{71} + 46362 q^{73} - 51399 q^{75} - 2548 q^{77} - 97424 q^{79} + 6561 q^{81} + 81228 q^{83} - 190068 q^{85} - 49662 q^{87} - 3182 q^{89} - 37730 q^{91} + 57024 q^{93} - 162808 q^{95} + 4914 q^{97} - 4212 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 94.0000 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.i 1
3.b odd 2 1 1008.6.a.a 1
4.b odd 2 1 21.6.a.c 1
12.b even 2 1 63.6.a.b 1
20.d odd 2 1 525.6.a.b 1
20.e even 4 2 525.6.d.c 2
28.d even 2 1 147.6.a.f 1
28.f even 6 2 147.6.e.d 2
28.g odd 6 2 147.6.e.c 2
84.h odd 2 1 441.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 4.b odd 2 1
63.6.a.b 1 12.b even 2 1
147.6.a.f 1 28.d even 2 1
147.6.e.c 2 28.g odd 6 2
147.6.e.d 2 28.f even 6 2
336.6.a.i 1 1.a even 1 1 trivial
441.6.a.c 1 84.h odd 2 1
525.6.a.b 1 20.d odd 2 1
525.6.d.c 2 20.e even 4 2
1008.6.a.a 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} - 94 \) Copy content Toggle raw display
\( T_{11} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T - 94 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 52 \) Copy content Toggle raw display
$13$ \( T + 770 \) Copy content Toggle raw display
$17$ \( T + 2022 \) Copy content Toggle raw display
$19$ \( T + 1732 \) Copy content Toggle raw display
$23$ \( T - 576 \) Copy content Toggle raw display
$29$ \( T - 5518 \) Copy content Toggle raw display
$31$ \( T + 6336 \) Copy content Toggle raw display
$37$ \( T + 7338 \) Copy content Toggle raw display
$41$ \( T + 3262 \) Copy content Toggle raw display
$43$ \( T + 5420 \) Copy content Toggle raw display
$47$ \( T + 864 \) Copy content Toggle raw display
$53$ \( T - 4182 \) Copy content Toggle raw display
$59$ \( T - 11220 \) Copy content Toggle raw display
$61$ \( T + 45602 \) Copy content Toggle raw display
$67$ \( T + 1396 \) Copy content Toggle raw display
$71$ \( T + 18720 \) Copy content Toggle raw display
$73$ \( T - 46362 \) Copy content Toggle raw display
$79$ \( T + 97424 \) Copy content Toggle raw display
$83$ \( T - 81228 \) Copy content Toggle raw display
$89$ \( T + 3182 \) Copy content Toggle raw display
$97$ \( T - 4914 \) Copy content Toggle raw display
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