Properties

Label 336.4.a.l
Level $336$
Weight $4$
Character orbit 336.a
Self dual yes
Analytic conductor $19.825$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 18q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 18q^{5} - 7q^{7} + 9q^{9} + 72q^{11} - 34q^{13} + 54q^{15} + 6q^{17} - 92q^{19} - 21q^{21} + 180q^{23} + 199q^{25} + 27q^{27} - 114q^{29} - 56q^{31} + 216q^{33} - 126q^{35} - 34q^{37} - 102q^{39} + 6q^{41} - 164q^{43} + 162q^{45} - 168q^{47} + 49q^{49} + 18q^{51} + 654q^{53} + 1296q^{55} - 276q^{57} + 492q^{59} - 250q^{61} - 63q^{63} - 612q^{65} + 124q^{67} + 540q^{69} - 36q^{71} + 1010q^{73} + 597q^{75} - 504q^{77} - 56q^{79} + 81q^{81} - 228q^{83} + 108q^{85} - 342q^{87} + 390q^{89} + 238q^{91} - 168q^{93} - 1656q^{95} - 70q^{97} + 648q^{99} + O(q^{100}) \)

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 18.0000 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.l 1
3.b odd 2 1 1008.4.a.b 1
4.b odd 2 1 42.4.a.a 1
7.b odd 2 1 2352.4.a.a 1
8.b even 2 1 1344.4.a.a 1
8.d odd 2 1 1344.4.a.o 1
12.b even 2 1 126.4.a.a 1
20.d odd 2 1 1050.4.a.g 1
20.e even 4 2 1050.4.g.a 2
28.d even 2 1 294.4.a.i 1
28.f even 6 2 294.4.e.b 2
28.g odd 6 2 294.4.e.c 2
84.h odd 2 1 882.4.a.g 1
84.j odd 6 2 882.4.g.o 2
84.n even 6 2 882.4.g.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 4.b odd 2 1
126.4.a.a 1 12.b even 2 1
294.4.a.i 1 28.d even 2 1
294.4.e.b 2 28.f even 6 2
294.4.e.c 2 28.g odd 6 2
336.4.a.l 1 1.a even 1 1 trivial
882.4.a.g 1 84.h odd 2 1
882.4.g.o 2 84.j odd 6 2
882.4.g.w 2 84.n even 6 2
1008.4.a.b 1 3.b odd 2 1
1050.4.a.g 1 20.d odd 2 1
1050.4.g.a 2 20.e even 4 2
1344.4.a.a 1 8.b even 2 1
1344.4.a.o 1 8.d odd 2 1
2352.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} - 18 \)
\( T_{11} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -18 + T \)
$7$ \( 7 + T \)
$11$ \( -72 + T \)
$13$ \( 34 + T \)
$17$ \( -6 + T \)
$19$ \( 92 + T \)
$23$ \( -180 + T \)
$29$ \( 114 + T \)
$31$ \( 56 + T \)
$37$ \( 34 + T \)
$41$ \( -6 + T \)
$43$ \( 164 + T \)
$47$ \( 168 + T \)
$53$ \( -654 + T \)
$59$ \( -492 + T \)
$61$ \( 250 + T \)
$67$ \( -124 + T \)
$71$ \( 36 + T \)
$73$ \( -1010 + T \)
$79$ \( 56 + T \)
$83$ \( 228 + T \)
$89$ \( -390 + T \)
$97$ \( 70 + T \)
show more
show less