Properties

Label 336.2.a.d
Level $336$
Weight $2$
Character orbit 336.a
Self dual yes
Analytic conductor $2.683$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.68297350792\)
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

\(f(q)\) \(=\) \( q + q^{3} - 2q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{5} + q^{7} + q^{9} + 4q^{11} + 6q^{13} - 2q^{15} + 2q^{17} + 4q^{19} + q^{21} - 8q^{23} - q^{25} + q^{27} - 2q^{29} + 4q^{33} - 2q^{35} - 10q^{37} + 6q^{39} - 6q^{41} + 4q^{43} - 2q^{45} + q^{49} + 2q^{51} + 6q^{53} - 8q^{55} + 4q^{57} - 4q^{59} + 6q^{61} + q^{63} - 12q^{65} - 4q^{67} - 8q^{69} - 8q^{71} + 10q^{73} - q^{75} + 4q^{77} + q^{81} + 4q^{83} - 4q^{85} - 2q^{87} - 6q^{89} + 6q^{91} - 8q^{95} - 14q^{97} + 4q^{99} + O(q^{100}) \)

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 1.00000 0 −2.00000 0 1.00000 0 1.00000 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 336.2.a.d 1
3.b odd 2 1 1008.2.a.j 1
4.b odd 2 1 42.2.a.a 1
5.b even 2 1 8400.2.a.k 1
7.b odd 2 1 2352.2.a.l 1
7.c even 3 2 2352.2.q.i 2
7.d odd 6 2 2352.2.q.n 2
8.b even 2 1 1344.2.a.i 1
8.d odd 2 1 1344.2.a.q 1
12.b even 2 1 126.2.a.a 1
16.e even 4 2 5376.2.c.e 2
16.f odd 4 2 5376.2.c.bc 2
20.d odd 2 1 1050.2.a.i 1
20.e even 4 2 1050.2.g.a 2
21.c even 2 1 7056.2.a.k 1
24.f even 2 1 4032.2.a.e 1
24.h odd 2 1 4032.2.a.m 1
28.d even 2 1 294.2.a.g 1
28.f even 6 2 294.2.e.a 2
28.g odd 6 2 294.2.e.c 2
36.f odd 6 2 1134.2.f.g 2
36.h even 6 2 1134.2.f.j 2
44.c even 2 1 5082.2.a.d 1
52.b odd 2 1 7098.2.a.f 1
56.e even 2 1 9408.2.a.n 1
56.h odd 2 1 9408.2.a.bw 1
60.h even 2 1 3150.2.a.bo 1
60.l odd 4 2 3150.2.g.r 2
84.h odd 2 1 882.2.a.b 1
84.j odd 6 2 882.2.g.j 2
84.n even 6 2 882.2.g.h 2
140.c even 2 1 7350.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 4.b odd 2 1
126.2.a.a 1 12.b even 2 1
294.2.a.g 1 28.d even 2 1
294.2.e.a 2 28.f even 6 2
294.2.e.c 2 28.g odd 6 2
336.2.a.d 1 1.a even 1 1 trivial
882.2.a.b 1 84.h odd 2 1
882.2.g.h 2 84.n even 6 2
882.2.g.j 2 84.j odd 6 2
1008.2.a.j 1 3.b odd 2 1
1050.2.a.i 1 20.d odd 2 1
1050.2.g.a 2 20.e even 4 2
1134.2.f.g 2 36.f odd 6 2
1134.2.f.j 2 36.h even 6 2
1344.2.a.i 1 8.b even 2 1
1344.2.a.q 1 8.d odd 2 1
2352.2.a.l 1 7.b odd 2 1
2352.2.q.i 2 7.c even 3 2
2352.2.q.n 2 7.d odd 6 2
3150.2.a.bo 1 60.h even 2 1
3150.2.g.r 2 60.l odd 4 2
4032.2.a.e 1 24.f even 2 1
4032.2.a.m 1 24.h odd 2 1
5082.2.a.d 1 44.c even 2 1
5376.2.c.e 2 16.e even 4 2
5376.2.c.bc 2 16.f odd 4 2
7056.2.a.k 1 21.c even 2 1
7098.2.a.f 1 52.b odd 2 1
7350.2.a.f 1 140.c even 2 1
8400.2.a.k 1 5.b even 2 1
9408.2.a.n 1 56.e even 2 1
9408.2.a.bw 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} + 2 \)
\( T_{11} - 4 \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 - T \)
$11$ \( 1 - 4 T + 11 T^{2} \)
$13$ \( 1 - 6 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 8 T + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 + 10 T + 37 T^{2} \)
$41$ \( 1 + 6 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 - 6 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 8 T + 71 T^{2} \)
$73$ \( 1 - 10 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 - 4 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 + 14 T + 97 T^{2} \)
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