Properties

Label 336.4.a.k
Level $336$
Weight $4$
Character orbit 336.a
Self dual yes
Analytic conductor $19.825$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 6q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 6q^{5} - 7q^{7} + 9q^{9} - 36q^{11} + 62q^{13} + 18q^{15} + 114q^{17} + 76q^{19} - 21q^{21} + 24q^{23} - 89q^{25} + 27q^{27} + 54q^{29} + 112q^{31} - 108q^{33} - 42q^{35} - 178q^{37} + 186q^{39} + 378q^{41} + 172q^{43} + 54q^{45} + 192q^{47} + 49q^{49} + 342q^{51} - 402q^{53} - 216q^{55} + 228q^{57} - 396q^{59} + 254q^{61} - 63q^{63} + 372q^{65} + 1012q^{67} + 72q^{69} - 840q^{71} + 890q^{73} - 267q^{75} + 252q^{77} - 80q^{79} + 81q^{81} + 108q^{83} + 684q^{85} + 162q^{87} - 1638q^{89} - 434q^{91} + 336q^{93} + 456q^{95} + 1010q^{97} - 324q^{99} + O(q^{100}) \)

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 3.00000 0 6.00000 0 −7.00000 0 9.00000 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 336.4.a.k 1
3.b odd 2 1 1008.4.a.h 1
4.b odd 2 1 84.4.a.a 1
7.b odd 2 1 2352.4.a.d 1
8.b even 2 1 1344.4.a.d 1
8.d odd 2 1 1344.4.a.q 1
12.b even 2 1 252.4.a.b 1
20.d odd 2 1 2100.4.a.l 1
20.e even 4 2 2100.4.k.j 2
28.d even 2 1 588.4.a.d 1
28.f even 6 2 588.4.i.c 2
28.g odd 6 2 588.4.i.f 2
84.h odd 2 1 1764.4.a.j 1
84.j odd 6 2 1764.4.k.f 2
84.n even 6 2 1764.4.k.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 4.b odd 2 1
252.4.a.b 1 12.b even 2 1
336.4.a.k 1 1.a even 1 1 trivial
588.4.a.d 1 28.d even 2 1
588.4.i.c 2 28.f even 6 2
588.4.i.f 2 28.g odd 6 2
1008.4.a.h 1 3.b odd 2 1
1344.4.a.d 1 8.b even 2 1
1344.4.a.q 1 8.d odd 2 1
1764.4.a.j 1 84.h odd 2 1
1764.4.k.f 2 84.j odd 6 2
1764.4.k.l 2 84.n even 6 2
2100.4.a.l 1 20.d odd 2 1
2100.4.k.j 2 20.e even 4 2
2352.4.a.d 1 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} - 6 \)
\( T_{11} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -6 + T \)
$7$ \( 7 + T \)
$11$ \( 36 + T \)
$13$ \( -62 + T \)
$17$ \( -114 + T \)
$19$ \( -76 + T \)
$23$ \( -24 + T \)
$29$ \( -54 + T \)
$31$ \( -112 + T \)
$37$ \( 178 + T \)
$41$ \( -378 + T \)
$43$ \( -172 + T \)
$47$ \( -192 + T \)
$53$ \( 402 + T \)
$59$ \( 396 + T \)
$61$ \( -254 + T \)
$67$ \( -1012 + T \)
$71$ \( 840 + T \)
$73$ \( -890 + T \)
$79$ \( 80 + T \)
$83$ \( -108 + T \)
$89$ \( 1638 + T \)
$97$ \( -1010 + T \)
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