Properties

Label 448.6.a.n
Level $448$
Weight $6$
Character orbit 448.a
Self dual yes
Analytic conductor $71.852$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.8519512762\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 14 q^{3} + 64 q^{5} + 49 q^{7} - 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 14 q^{3} + 64 q^{5} + 49 q^{7} - 47 q^{9} - 420 q^{11} - 860 q^{13} + 896 q^{15} + 830 q^{17} + 490 q^{19} + 686 q^{21} - 4872 q^{23} + 971 q^{25} - 4060 q^{27} - 8754 q^{29} - 5628 q^{31} - 5880 q^{33} + 3136 q^{35} - 1434 q^{37} - 12040 q^{39} - 9258 q^{41} + 14756 q^{43} - 3008 q^{45} + 10108 q^{47} + 2401 q^{49} + 11620 q^{51} + 23058 q^{53} - 26880 q^{55} + 6860 q^{57} - 13734 q^{59} + 25352 q^{61} - 2303 q^{63} - 55040 q^{65} + 19768 q^{67} - 68208 q^{69} - 1792 q^{71} + 37914 q^{73} + 13594 q^{75} - 20580 q^{77} - 95984 q^{79} - 45419 q^{81} + 88242 q^{83} + 53120 q^{85} - 122556 q^{87} + 43762 q^{89} - 42140 q^{91} - 78792 q^{93} + 31360 q^{95} + 65790 q^{97} + 19740 q^{99}+O(q^{100}) \) 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 14.0000 0 64.0000 0 49.0000 0 −47.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

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 448.6.a.n 1
4.b odd 2 1 448.6.a.d 1
8.b even 2 1 224.6.a.a 1
8.d odd 2 1 224.6.a.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.a 1 8.b even 2 1
224.6.a.b yes 1 8.d odd 2 1
448.6.a.d 1 4.b odd 2 1
448.6.a.n 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3} - 14 \) Copy content Toggle raw display
\( T_{5} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 14 \) Copy content Toggle raw display
$5$ \( T - 64 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 420 \) Copy content Toggle raw display
$13$ \( T + 860 \) Copy content Toggle raw display
$17$ \( T - 830 \) Copy content Toggle raw display
$19$ \( T - 490 \) Copy content Toggle raw display
$23$ \( T + 4872 \) Copy content Toggle raw display
$29$ \( T + 8754 \) Copy content Toggle raw display
$31$ \( T + 5628 \) Copy content Toggle raw display
$37$ \( T + 1434 \) Copy content Toggle raw display
$41$ \( T + 9258 \) Copy content Toggle raw display
$43$ \( T - 14756 \) Copy content Toggle raw display
$47$ \( T - 10108 \) Copy content Toggle raw display
$53$ \( T - 23058 \) Copy content Toggle raw display
$59$ \( T + 13734 \) Copy content Toggle raw display
$61$ \( T - 25352 \) Copy content Toggle raw display
$67$ \( T - 19768 \) Copy content Toggle raw display
$71$ \( T + 1792 \) Copy content Toggle raw display
$73$ \( T - 37914 \) Copy content Toggle raw display
$79$ \( T + 95984 \) Copy content Toggle raw display
$83$ \( T - 88242 \) Copy content Toggle raw display
$89$ \( T - 43762 \) Copy content Toggle raw display
$97$ \( T - 65790 \) Copy content Toggle raw display
show more
show less