Properties

Label 1344.2.a.i
Level $1344$
Weight $2$
Character orbit 1344.a
Self dual yes
Analytic conductor $10.732$
Analytic rank $1$
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: \(1\)
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

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

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.i 1
3.b odd 2 1 4032.2.a.m 1
4.b odd 2 1 1344.2.a.q 1
7.b odd 2 1 9408.2.a.bw 1
8.b even 2 1 336.2.a.d 1
8.d odd 2 1 42.2.a.a 1
12.b even 2 1 4032.2.a.e 1
16.e even 4 2 5376.2.c.e 2
16.f odd 4 2 5376.2.c.bc 2
24.f even 2 1 126.2.a.a 1
24.h odd 2 1 1008.2.a.j 1
28.d even 2 1 9408.2.a.n 1
40.e odd 2 1 1050.2.a.i 1
40.f even 2 1 8400.2.a.k 1
40.k even 4 2 1050.2.g.a 2
56.e even 2 1 294.2.a.g 1
56.h odd 2 1 2352.2.a.l 1
56.j odd 6 2 2352.2.q.n 2
56.k odd 6 2 294.2.e.c 2
56.m even 6 2 294.2.e.a 2
56.p even 6 2 2352.2.q.i 2
72.l even 6 2 1134.2.f.j 2
72.p odd 6 2 1134.2.f.g 2
88.g even 2 1 5082.2.a.d 1
104.h odd 2 1 7098.2.a.f 1
120.m even 2 1 3150.2.a.bo 1
120.q odd 4 2 3150.2.g.r 2
168.e odd 2 1 882.2.a.b 1
168.i even 2 1 7056.2.a.k 1
168.v even 6 2 882.2.g.h 2
168.be odd 6 2 882.2.g.j 2
280.n 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 8.d odd 2 1
126.2.a.a 1 24.f even 2 1
294.2.a.g 1 56.e even 2 1
294.2.e.a 2 56.m even 6 2
294.2.e.c 2 56.k odd 6 2
336.2.a.d 1 8.b even 2 1
882.2.a.b 1 168.e odd 2 1
882.2.g.h 2 168.v even 6 2
882.2.g.j 2 168.be odd 6 2
1008.2.a.j 1 24.h odd 2 1
1050.2.a.i 1 40.e odd 2 1
1050.2.g.a 2 40.k even 4 2
1134.2.f.g 2 72.p odd 6 2
1134.2.f.j 2 72.l even 6 2
1344.2.a.i 1 1.a even 1 1 trivial
1344.2.a.q 1 4.b odd 2 1
2352.2.a.l 1 56.h odd 2 1
2352.2.q.i 2 56.p even 6 2
2352.2.q.n 2 56.j odd 6 2
3150.2.a.bo 1 120.m even 2 1
3150.2.g.r 2 120.q odd 4 2
4032.2.a.e 1 12.b even 2 1
4032.2.a.m 1 3.b odd 2 1
5082.2.a.d 1 88.g 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 1 168.i even 2 1
7098.2.a.f 1 104.h odd 2 1
7350.2.a.f 1 280.n even 2 1
8400.2.a.k 1 40.f even 2 1
9408.2.a.n 1 28.d even 2 1
9408.2.a.bw 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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