Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
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

\(f(q)\) \(=\) \( q + 3q^{3} - 4q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 4q^{5} + 7q^{7} + 9q^{9} - 62q^{11} - 62q^{13} - 12q^{15} + 84q^{17} - 100q^{19} + 21q^{21} + 42q^{23} - 109q^{25} + 27q^{27} - 10q^{29} + 48q^{31} - 186q^{33} - 28q^{35} - 246q^{37} - 186q^{39} - 248q^{41} - 68q^{43} - 36q^{45} - 324q^{47} + 49q^{49} + 252q^{51} + 258q^{53} + 248q^{55} - 300q^{57} - 120q^{59} + 622q^{61} + 63q^{63} + 248q^{65} - 904q^{67} + 126q^{69} + 678q^{71} - 642q^{73} - 327q^{75} - 434q^{77} - 740q^{79} + 81q^{81} - 468q^{83} - 336q^{85} - 30q^{87} + 200q^{89} - 434q^{91} + 144q^{93} + 400q^{95} - 1266q^{97} - 558q^{99} + O(q^{100}) \)

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 −4.00000 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.h 1
3.b odd 2 1 1008.4.a.m 1
4.b odd 2 1 21.4.a.b 1
7.b odd 2 1 2352.4.a.l 1
8.b even 2 1 1344.4.a.i 1
8.d odd 2 1 1344.4.a.w 1
12.b even 2 1 63.4.a.a 1
20.d odd 2 1 525.4.a.b 1
20.e even 4 2 525.4.d.b 2
28.d even 2 1 147.4.a.g 1
28.f even 6 2 147.4.e.b 2
28.g odd 6 2 147.4.e.c 2
60.h even 2 1 1575.4.a.k 1
84.h odd 2 1 441.4.a.b 1
84.j odd 6 2 441.4.e.n 2
84.n even 6 2 441.4.e.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 4.b odd 2 1
63.4.a.a 1 12.b even 2 1
147.4.a.g 1 28.d even 2 1
147.4.e.b 2 28.f even 6 2
147.4.e.c 2 28.g odd 6 2
336.4.a.h 1 1.a even 1 1 trivial
441.4.a.b 1 84.h odd 2 1
441.4.e.m 2 84.n even 6 2
441.4.e.n 2 84.j odd 6 2
525.4.a.b 1 20.d odd 2 1
525.4.d.b 2 20.e even 4 2
1008.4.a.m 1 3.b odd 2 1
1344.4.a.i 1 8.b even 2 1
1344.4.a.w 1 8.d odd 2 1
1575.4.a.k 1 60.h even 2 1
2352.4.a.l 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} + 4 \)
\( T_{11} + 62 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 4 + T \)
$7$ \( -7 + T \)
$11$ \( 62 + T \)
$13$ \( 62 + T \)
$17$ \( -84 + T \)
$19$ \( 100 + T \)
$23$ \( -42 + T \)
$29$ \( 10 + T \)
$31$ \( -48 + T \)
$37$ \( 246 + T \)
$41$ \( 248 + T \)
$43$ \( 68 + T \)
$47$ \( 324 + T \)
$53$ \( -258 + T \)
$59$ \( 120 + T \)
$61$ \( -622 + T \)
$67$ \( 904 + T \)
$71$ \( -678 + T \)
$73$ \( 642 + T \)
$79$ \( 740 + T \)
$83$ \( 468 + T \)
$89$ \( -200 + T \)
$97$ \( 1266 + T \)
show more
show less