Properties

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

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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 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 14 q^{3} + 56 q^{5} + 49 q^{7} - 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 14 q^{3} + 56 q^{5} + 49 q^{7} - 47 q^{9} + 232 q^{11} + 140 q^{13} - 784 q^{15} - 1722 q^{17} - 98 q^{19} - 686 q^{21} - 1824 q^{23} + 11 q^{25} + 4060 q^{27} - 3418 q^{29} + 7644 q^{31} - 3248 q^{33} + 2744 q^{35} + 10398 q^{37} - 1960 q^{39} - 17962 q^{41} + 10880 q^{43} - 2632 q^{45} - 9324 q^{47} + 2401 q^{49} + 24108 q^{51} - 2262 q^{53} + 12992 q^{55} + 1372 q^{57} - 2730 q^{59} - 25648 q^{61} - 2303 q^{63} + 7840 q^{65} - 48404 q^{67} + 25536 q^{69} + 58560 q^{71} + 68082 q^{73} - 154 q^{75} + 11368 q^{77} - 31784 q^{79} - 45419 q^{81} - 20538 q^{83} - 96432 q^{85} + 47852 q^{87} - 50582 q^{89} + 6860 q^{91} - 107016 q^{93} - 5488 q^{95} - 58506 q^{97} - 10904 q^{99}+O(q^{100}) \) 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 56.0000 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.c 1
4.b odd 2 1 448.6.a.m 1
8.b even 2 1 112.6.a.g 1
8.d odd 2 1 7.6.a.a 1
24.f even 2 1 63.6.a.e 1
24.h odd 2 1 1008.6.a.y 1
40.e odd 2 1 175.6.a.b 1
40.k even 4 2 175.6.b.a 2
56.e even 2 1 49.6.a.a 1
56.h odd 2 1 784.6.a.c 1
56.k odd 6 2 49.6.c.c 2
56.m even 6 2 49.6.c.b 2
88.g even 2 1 847.6.a.b 1
168.e odd 2 1 441.6.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 8.d odd 2 1
49.6.a.a 1 56.e even 2 1
49.6.c.b 2 56.m even 6 2
49.6.c.c 2 56.k odd 6 2
63.6.a.e 1 24.f even 2 1
112.6.a.g 1 8.b even 2 1
175.6.a.b 1 40.e odd 2 1
175.6.b.a 2 40.k even 4 2
441.6.a.k 1 168.e odd 2 1
448.6.a.c 1 1.a even 1 1 trivial
448.6.a.m 1 4.b odd 2 1
784.6.a.c 1 56.h odd 2 1
847.6.a.b 1 88.g even 2 1
1008.6.a.y 1 24.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3} + 14 \) Copy content Toggle raw display
\( T_{5} - 56 \) 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
$5$ \( T - 56 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 232 \) Copy content Toggle raw display
$13$ \( T - 140 \) Copy content Toggle raw display
$17$ \( T + 1722 \) Copy content Toggle raw display
$19$ \( T + 98 \) Copy content Toggle raw display
$23$ \( T + 1824 \) Copy content Toggle raw display
$29$ \( T + 3418 \) Copy content Toggle raw display
$31$ \( T - 7644 \) Copy content Toggle raw display
$37$ \( T - 10398 \) Copy content Toggle raw display
$41$ \( T + 17962 \) Copy content Toggle raw display
$43$ \( T - 10880 \) Copy content Toggle raw display
$47$ \( T + 9324 \) Copy content Toggle raw display
$53$ \( T + 2262 \) Copy content Toggle raw display
$59$ \( T + 2730 \) Copy content Toggle raw display
$61$ \( T + 25648 \) Copy content Toggle raw display
$67$ \( T + 48404 \) Copy content Toggle raw display
$71$ \( T - 58560 \) Copy content Toggle raw display
$73$ \( T - 68082 \) Copy content Toggle raw display
$79$ \( T + 31784 \) Copy content Toggle raw display
$83$ \( T + 20538 \) Copy content Toggle raw display
$89$ \( T + 50582 \) Copy content Toggle raw display
$97$ \( T + 58506 \) Copy content Toggle raw display
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