Properties

Label 1344.4.a.o
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(0\)
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

\(f(q)\) \(=\) \( q + 3q^{3} - 18q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 18q^{5} + 7q^{7} + 9q^{9} + 72q^{11} + 34q^{13} - 54q^{15} + 6q^{17} - 92q^{19} + 21q^{21} - 180q^{23} + 199q^{25} + 27q^{27} + 114q^{29} + 56q^{31} + 216q^{33} - 126q^{35} + 34q^{37} + 102q^{39} + 6q^{41} - 164q^{43} - 162q^{45} + 168q^{47} + 49q^{49} + 18q^{51} - 654q^{53} - 1296q^{55} - 276q^{57} + 492q^{59} + 250q^{61} + 63q^{63} - 612q^{65} + 124q^{67} - 540q^{69} + 36q^{71} + 1010q^{73} + 597q^{75} + 504q^{77} + 56q^{79} + 81q^{81} - 228q^{83} - 108q^{85} + 342q^{87} + 390q^{89} + 238q^{91} + 168q^{93} + 1656q^{95} - 70q^{97} + 648q^{99} + O(q^{100}) \)

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 −18.0000 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.o 1
4.b odd 2 1 1344.4.a.a 1
8.b even 2 1 42.4.a.a 1
8.d odd 2 1 336.4.a.l 1
24.f even 2 1 1008.4.a.b 1
24.h odd 2 1 126.4.a.a 1
40.f even 2 1 1050.4.a.g 1
40.i odd 4 2 1050.4.g.a 2
56.e even 2 1 2352.4.a.a 1
56.h odd 2 1 294.4.a.i 1
56.j odd 6 2 294.4.e.b 2
56.p even 6 2 294.4.e.c 2
168.i even 2 1 882.4.a.g 1
168.s odd 6 2 882.4.g.w 2
168.ba even 6 2 882.4.g.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 8.b even 2 1
126.4.a.a 1 24.h odd 2 1
294.4.a.i 1 56.h odd 2 1
294.4.e.b 2 56.j odd 6 2
294.4.e.c 2 56.p even 6 2
336.4.a.l 1 8.d odd 2 1
882.4.a.g 1 168.i even 2 1
882.4.g.o 2 168.ba even 6 2
882.4.g.w 2 168.s odd 6 2
1008.4.a.b 1 24.f even 2 1
1050.4.a.g 1 40.f even 2 1
1050.4.g.a 2 40.i odd 4 2
1344.4.a.a 1 4.b odd 2 1
1344.4.a.o 1 1.a even 1 1 trivial
2352.4.a.a 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} + 18 \)
\( T_{11} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 18 + T \)
$7$ \( -7 + T \)
$11$ \( -72 + T \)
$13$ \( -34 + T \)
$17$ \( -6 + T \)
$19$ \( 92 + T \)
$23$ \( 180 + T \)
$29$ \( -114 + T \)
$31$ \( -56 + T \)
$37$ \( -34 + T \)
$41$ \( -6 + T \)
$43$ \( 164 + T \)
$47$ \( -168 + T \)
$53$ \( 654 + T \)
$59$ \( -492 + T \)
$61$ \( -250 + T \)
$67$ \( -124 + T \)
$71$ \( -36 + T \)
$73$ \( -1010 + T \)
$79$ \( -56 + T \)
$83$ \( 228 + T \)
$89$ \( -390 + T \)
$97$ \( 70 + T \)
show more
show less