Properties

Label 336.2.a.c
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 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 2q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + 2q^{5} + q^{7} + q^{9} - 2q^{13} - 2q^{15} + 6q^{17} + 4q^{19} - q^{21} + 4q^{23} - q^{25} - q^{27} + 6q^{29} + 8q^{31} + 2q^{35} - 10q^{37} + 2q^{39} - 10q^{41} - 12q^{43} + 2q^{45} + 8q^{47} + q^{49} - 6q^{51} + 6q^{53} - 4q^{57} - 4q^{59} - 10q^{61} + q^{63} - 4q^{65} - 12q^{67} - 4q^{69} - 4q^{71} + 2q^{73} + q^{75} - 8q^{79} + q^{81} - 4q^{83} + 12q^{85} - 6q^{87} + 6q^{89} - 2q^{91} - 8q^{93} + 8q^{95} + 10q^{97} + O(q^{100}) \)

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:

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.c 1
3.b odd 2 1 1008.2.a.e 1
4.b odd 2 1 168.2.a.b 1
5.b even 2 1 8400.2.a.bx 1
7.b odd 2 1 2352.2.a.q 1
7.c even 3 2 2352.2.q.o 2
7.d odd 6 2 2352.2.q.j 2
8.b even 2 1 1344.2.a.n 1
8.d odd 2 1 1344.2.a.c 1
12.b even 2 1 504.2.a.b 1
16.e even 4 2 5376.2.c.f 2
16.f odd 4 2 5376.2.c.bd 2
20.d odd 2 1 4200.2.a.i 1
20.e even 4 2 4200.2.t.m 2
21.c even 2 1 7056.2.a.br 1
24.f even 2 1 4032.2.a.be 1
24.h odd 2 1 4032.2.a.bj 1
28.d even 2 1 1176.2.a.a 1
28.f even 6 2 1176.2.q.j 2
28.g odd 6 2 1176.2.q.b 2
56.e even 2 1 9408.2.a.cy 1
56.h odd 2 1 9408.2.a.bc 1
84.h odd 2 1 3528.2.a.w 1
84.j odd 6 2 3528.2.s.h 2
84.n even 6 2 3528.2.s.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.b 1 4.b odd 2 1
336.2.a.c 1 1.a even 1 1 trivial
504.2.a.b 1 12.b even 2 1
1008.2.a.e 1 3.b odd 2 1
1176.2.a.a 1 28.d even 2 1
1176.2.q.b 2 28.g odd 6 2
1176.2.q.j 2 28.f even 6 2
1344.2.a.c 1 8.d odd 2 1
1344.2.a.n 1 8.b even 2 1
2352.2.a.q 1 7.b odd 2 1
2352.2.q.j 2 7.d odd 6 2
2352.2.q.o 2 7.c even 3 2
3528.2.a.w 1 84.h odd 2 1
3528.2.s.h 2 84.j odd 6 2
3528.2.s.v 2 84.n even 6 2
4032.2.a.be 1 24.f even 2 1
4032.2.a.bj 1 24.h odd 2 1
4200.2.a.i 1 20.d odd 2 1
4200.2.t.m 2 20.e even 4 2
5376.2.c.f 2 16.e even 4 2
5376.2.c.bd 2 16.f odd 4 2
7056.2.a.br 1 21.c even 2 1
8400.2.a.bx 1 5.b even 2 1
9408.2.a.bc 1 56.h odd 2 1
9408.2.a.cy 1 56.e even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(336))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 - T \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 - 6 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 + 10 T + 37 T^{2} \)
$41$ \( 1 + 10 T + 41 T^{2} \)
$43$ \( 1 + 12 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 + 4 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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