Properties

Label 336.4.a.e
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

\(f(q)\) \(=\) \( q - 3q^{3} + 14q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 14q^{5} + 7q^{7} + 9q^{9} - 4q^{11} + 54q^{13} - 42q^{15} - 14q^{17} - 92q^{19} - 21q^{21} + 152q^{23} + 71q^{25} - 27q^{27} - 106q^{29} + 144q^{31} + 12q^{33} + 98q^{35} + 158q^{37} - 162q^{39} - 390q^{41} + 508q^{43} + 126q^{45} + 528q^{47} + 49q^{49} + 42q^{51} + 606q^{53} - 56q^{55} + 276q^{57} + 364q^{59} + 678q^{61} + 63q^{63} + 756q^{65} - 844q^{67} - 456q^{69} + 8q^{71} - 422q^{73} - 213q^{75} - 28q^{77} - 384q^{79} + 81q^{81} + 548q^{83} - 196q^{85} + 318q^{87} + 1194q^{89} + 378q^{91} - 432q^{93} - 1288q^{95} - 1502q^{97} - 36q^{99} + O(q^{100}) \)

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 14.0000 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.e 1
3.b odd 2 1 1008.4.a.d 1
4.b odd 2 1 84.4.a.b 1
7.b odd 2 1 2352.4.a.v 1
8.b even 2 1 1344.4.a.p 1
8.d odd 2 1 1344.4.a.b 1
12.b even 2 1 252.4.a.a 1
20.d odd 2 1 2100.4.a.g 1
20.e even 4 2 2100.4.k.g 2
28.d even 2 1 588.4.a.a 1
28.f even 6 2 588.4.i.h 2
28.g odd 6 2 588.4.i.a 2
84.h odd 2 1 1764.4.a.l 1
84.j odd 6 2 1764.4.k.c 2
84.n even 6 2 1764.4.k.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 4.b odd 2 1
252.4.a.a 1 12.b even 2 1
336.4.a.e 1 1.a even 1 1 trivial
588.4.a.a 1 28.d even 2 1
588.4.i.a 2 28.g odd 6 2
588.4.i.h 2 28.f even 6 2
1008.4.a.d 1 3.b odd 2 1
1344.4.a.b 1 8.d odd 2 1
1344.4.a.p 1 8.b even 2 1
1764.4.a.l 1 84.h odd 2 1
1764.4.k.c 2 84.j odd 6 2
1764.4.k.n 2 84.n even 6 2
2100.4.a.g 1 20.d odd 2 1
2100.4.k.g 2 20.e even 4 2
2352.4.a.v 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} - 14 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( -14 + T \)
$7$ \( -7 + T \)
$11$ \( 4 + T \)
$13$ \( -54 + T \)
$17$ \( 14 + T \)
$19$ \( 92 + T \)
$23$ \( -152 + T \)
$29$ \( 106 + T \)
$31$ \( -144 + T \)
$37$ \( -158 + T \)
$41$ \( 390 + T \)
$43$ \( -508 + T \)
$47$ \( -528 + T \)
$53$ \( -606 + T \)
$59$ \( -364 + T \)
$61$ \( -678 + T \)
$67$ \( 844 + T \)
$71$ \( -8 + T \)
$73$ \( 422 + T \)
$79$ \( 384 + T \)
$83$ \( -548 + T \)
$89$ \( -1194 + T \)
$97$ \( 1502 + T \)
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