Properties

Label 1344.4.a.v
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 3q^{3} + 2q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 2q^{5} + 7q^{7} + 9q^{9} + 52q^{11} - 86q^{13} + 6q^{15} - 30q^{17} - 4q^{19} + 21q^{21} - 120q^{23} - 121q^{25} + 27q^{27} - 246q^{29} - 80q^{31} + 156q^{33} + 14q^{35} + 290q^{37} - 258q^{39} - 374q^{41} + 164q^{43} + 18q^{45} - 464q^{47} + 49q^{49} - 90q^{51} + 162q^{53} + 104q^{55} - 12q^{57} + 180q^{59} + 666q^{61} + 63q^{63} - 172q^{65} - 628q^{67} - 360q^{69} - 296q^{71} - 518q^{73} - 363q^{75} + 364q^{77} + 1184q^{79} + 81q^{81} + 220q^{83} - 60q^{85} - 738q^{87} - 774q^{89} - 602q^{91} - 240q^{93} - 8q^{95} - 1086q^{97} + 468q^{99} + O(q^{100}) \)

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 2.00000 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 1344.4.a.v 1
4.b odd 2 1 1344.4.a.g 1
8.b even 2 1 336.4.a.c 1
8.d odd 2 1 168.4.a.f 1
24.f even 2 1 504.4.a.c 1
24.h odd 2 1 1008.4.a.l 1
56.e even 2 1 1176.4.a.e 1
56.h odd 2 1 2352.4.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.f 1 8.d odd 2 1
336.4.a.c 1 8.b even 2 1
504.4.a.c 1 24.f even 2 1
1008.4.a.l 1 24.h odd 2 1
1176.4.a.e 1 56.e even 2 1
1344.4.a.g 1 4.b odd 2 1
1344.4.a.v 1 1.a even 1 1 trivial
2352.4.a.bb 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} - 2 \)
\( T_{11} - 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -2 + T \)
$7$ \( -7 + T \)
$11$ \( -52 + T \)
$13$ \( 86 + T \)
$17$ \( 30 + T \)
$19$ \( 4 + T \)
$23$ \( 120 + T \)
$29$ \( 246 + T \)
$31$ \( 80 + T \)
$37$ \( -290 + T \)
$41$ \( 374 + T \)
$43$ \( -164 + T \)
$47$ \( 464 + T \)
$53$ \( -162 + T \)
$59$ \( -180 + T \)
$61$ \( -666 + T \)
$67$ \( 628 + T \)
$71$ \( 296 + T \)
$73$ \( 518 + T \)
$79$ \( -1184 + T \)
$83$ \( -220 + T \)
$89$ \( 774 + T \)
$97$ \( 1086 + T \)
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