Properties

Label 1344.2.a.s
Level $1344$
Weight $2$
Character orbit 1344.a
Self dual yes
Analytic conductor $10.732$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.7318940317\)
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

\(f(q)\) \(=\) \( q + q^{3} + 2q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 2q^{5} + q^{7} + q^{9} + 4q^{11} + 2q^{13} + 2q^{15} - 6q^{17} + 4q^{19} + q^{21} - q^{25} + q^{27} + 2q^{29} + 4q^{33} + 2q^{35} - 6q^{37} + 2q^{39} + 2q^{41} - 4q^{43} + 2q^{45} + q^{49} - 6q^{51} - 6q^{53} + 8q^{55} + 4q^{57} + 12q^{59} + 2q^{61} + q^{63} + 4q^{65} + 4q^{67} - 6q^{73} - q^{75} + 4q^{77} + 16q^{79} + q^{81} - 12q^{83} - 12q^{85} + 2q^{87} - 14q^{89} + 2q^{91} + 8q^{95} + 18q^{97} + 4q^{99} + O(q^{100}) \)

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 1344.2.a.s 1
3.b odd 2 1 4032.2.a.k 1
4.b odd 2 1 1344.2.a.g 1
7.b odd 2 1 9408.2.a.m 1
8.b even 2 1 336.2.a.a 1
8.d odd 2 1 21.2.a.a 1
12.b even 2 1 4032.2.a.h 1
16.e even 4 2 5376.2.c.l 2
16.f odd 4 2 5376.2.c.r 2
24.f even 2 1 63.2.a.a 1
24.h odd 2 1 1008.2.a.l 1
28.d even 2 1 9408.2.a.bv 1
40.e odd 2 1 525.2.a.d 1
40.f even 2 1 8400.2.a.bn 1
40.k even 4 2 525.2.d.a 2
56.e even 2 1 147.2.a.a 1
56.h odd 2 1 2352.2.a.v 1
56.j odd 6 2 2352.2.q.e 2
56.k odd 6 2 147.2.e.b 2
56.m even 6 2 147.2.e.c 2
56.p even 6 2 2352.2.q.x 2
72.l even 6 2 567.2.f.b 2
72.p odd 6 2 567.2.f.g 2
88.g even 2 1 2541.2.a.j 1
104.h odd 2 1 3549.2.a.c 1
120.m even 2 1 1575.2.a.c 1
120.q odd 4 2 1575.2.d.a 2
136.e odd 2 1 6069.2.a.b 1
152.b even 2 1 7581.2.a.d 1
168.e odd 2 1 441.2.a.f 1
168.i even 2 1 7056.2.a.p 1
168.v even 6 2 441.2.e.a 2
168.be odd 6 2 441.2.e.b 2
264.p odd 2 1 7623.2.a.g 1
280.n even 2 1 3675.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 8.d odd 2 1
63.2.a.a 1 24.f even 2 1
147.2.a.a 1 56.e even 2 1
147.2.e.b 2 56.k odd 6 2
147.2.e.c 2 56.m even 6 2
336.2.a.a 1 8.b even 2 1
441.2.a.f 1 168.e odd 2 1
441.2.e.a 2 168.v even 6 2
441.2.e.b 2 168.be odd 6 2
525.2.a.d 1 40.e odd 2 1
525.2.d.a 2 40.k even 4 2
567.2.f.b 2 72.l even 6 2
567.2.f.g 2 72.p odd 6 2
1008.2.a.l 1 24.h odd 2 1
1344.2.a.g 1 4.b odd 2 1
1344.2.a.s 1 1.a even 1 1 trivial
1575.2.a.c 1 120.m even 2 1
1575.2.d.a 2 120.q odd 4 2
2352.2.a.v 1 56.h odd 2 1
2352.2.q.e 2 56.j odd 6 2
2352.2.q.x 2 56.p even 6 2
2541.2.a.j 1 88.g even 2 1
3549.2.a.c 1 104.h odd 2 1
3675.2.a.n 1 280.n even 2 1
4032.2.a.h 1 12.b even 2 1
4032.2.a.k 1 3.b odd 2 1
5376.2.c.l 2 16.e even 4 2
5376.2.c.r 2 16.f odd 4 2
6069.2.a.b 1 136.e odd 2 1
7056.2.a.p 1 168.i even 2 1
7581.2.a.d 1 152.b even 2 1
7623.2.a.g 1 264.p odd 2 1
8400.2.a.bn 1 40.f even 2 1
9408.2.a.m 1 7.b odd 2 1
9408.2.a.bv 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} - 2 \)
\( T_{11} - 4 \)
\( T_{13} - 2 \)
\( T_{19} - 4 \)