Properties

Label 336.4.a.f
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 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 18q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 18q^{5} - 7q^{7} + 9q^{9} + 36q^{11} - 34q^{13} - 54q^{15} + 42q^{17} + 124q^{19} - 21q^{21} + 199q^{25} + 27q^{27} + 102q^{29} + 160q^{31} + 108q^{33} + 126q^{35} + 398q^{37} - 102q^{39} - 318q^{41} + 268q^{43} - 162q^{45} - 240q^{47} + 49q^{49} + 126q^{51} - 498q^{53} - 648q^{55} + 372q^{57} + 132q^{59} + 398q^{61} - 63q^{63} + 612q^{65} - 92q^{67} + 720q^{71} - 502q^{73} + 597q^{75} - 252q^{77} + 1024q^{79} + 81q^{81} + 204q^{83} - 756q^{85} + 306q^{87} + 354q^{89} + 238q^{91} + 480q^{93} - 2232q^{95} - 286q^{97} + 324q^{99} + O(q^{100}) \)

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 −18.0000 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.f 1
3.b odd 2 1 1008.4.a.v 1
4.b odd 2 1 21.4.a.a 1
7.b odd 2 1 2352.4.a.r 1
8.b even 2 1 1344.4.a.n 1
8.d odd 2 1 1344.4.a.ba 1
12.b even 2 1 63.4.a.c 1
20.d odd 2 1 525.4.a.g 1
20.e even 4 2 525.4.d.c 2
28.d even 2 1 147.4.a.c 1
28.f even 6 2 147.4.e.g 2
28.g odd 6 2 147.4.e.i 2
60.h even 2 1 1575.4.a.b 1
84.h odd 2 1 441.4.a.j 1
84.j odd 6 2 441.4.e.d 2
84.n even 6 2 441.4.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 4.b odd 2 1
63.4.a.c 1 12.b even 2 1
147.4.a.c 1 28.d even 2 1
147.4.e.g 2 28.f even 6 2
147.4.e.i 2 28.g odd 6 2
336.4.a.f 1 1.a even 1 1 trivial
441.4.a.j 1 84.h odd 2 1
441.4.e.b 2 84.n even 6 2
441.4.e.d 2 84.j odd 6 2
525.4.a.g 1 20.d odd 2 1
525.4.d.c 2 20.e even 4 2
1008.4.a.v 1 3.b odd 2 1
1344.4.a.n 1 8.b even 2 1
1344.4.a.ba 1 8.d odd 2 1
1575.4.a.b 1 60.h even 2 1
2352.4.a.r 1 7.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} + 18 \)
\( T_{11} - 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T \)
$5$ \( 1 + 18 T + 125 T^{2} \)
$7$ \( 1 + 7 T \)
$11$ \( 1 - 36 T + 1331 T^{2} \)
$13$ \( 1 + 34 T + 2197 T^{2} \)
$17$ \( 1 - 42 T + 4913 T^{2} \)
$19$ \( 1 - 124 T + 6859 T^{2} \)
$23$ \( 1 + 12167 T^{2} \)
$29$ \( 1 - 102 T + 24389 T^{2} \)
$31$ \( 1 - 160 T + 29791 T^{2} \)
$37$ \( 1 - 398 T + 50653 T^{2} \)
$41$ \( 1 + 318 T + 68921 T^{2} \)
$43$ \( 1 - 268 T + 79507 T^{2} \)
$47$ \( 1 + 240 T + 103823 T^{2} \)
$53$ \( 1 + 498 T + 148877 T^{2} \)
$59$ \( 1 - 132 T + 205379 T^{2} \)
$61$ \( 1 - 398 T + 226981 T^{2} \)
$67$ \( 1 + 92 T + 300763 T^{2} \)
$71$ \( 1 - 720 T + 357911 T^{2} \)
$73$ \( 1 + 502 T + 389017 T^{2} \)
$79$ \( 1 - 1024 T + 493039 T^{2} \)
$83$ \( 1 - 204 T + 571787 T^{2} \)
$89$ \( 1 - 354 T + 704969 T^{2} \)
$97$ \( 1 + 286 T + 912673 T^{2} \)
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