Properties

Label 448.6.a.l
Level $448$
Weight $6$
Character orbit 448.a
Self dual yes
Analytic conductor $71.852$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 10 q^{3} - 84 q^{5} - 49 q^{7} - 143 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 10 q^{3} - 84 q^{5} - 49 q^{7} - 143 q^{9} - 336 q^{11} - 584 q^{13} - 840 q^{15} - 1458 q^{17} + 470 q^{19} - 490 q^{21} + 4200 q^{23} + 3931 q^{25} - 3860 q^{27} - 4866 q^{29} + 7372 q^{31} - 3360 q^{33} + 4116 q^{35} - 14330 q^{37} - 5840 q^{39} + 6222 q^{41} + 3704 q^{43} + 12012 q^{45} + 1812 q^{47} + 2401 q^{49} - 14580 q^{51} + 37242 q^{53} + 28224 q^{55} + 4700 q^{57} + 34302 q^{59} - 24476 q^{61} + 7007 q^{63} + 49056 q^{65} - 17452 q^{67} + 42000 q^{69} - 28224 q^{71} + 3602 q^{73} + 39310 q^{75} + 16464 q^{77} - 42872 q^{79} - 3851 q^{81} - 35202 q^{83} + 122472 q^{85} - 48660 q^{87} + 26730 q^{89} + 28616 q^{91} + 73720 q^{93} - 39480 q^{95} - 16978 q^{97} + 48048 q^{99}+O(q^{100}) \) 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 10.0000 0 −84.0000 0 −49.0000 0 −143.000 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.l 1
4.b odd 2 1 448.6.a.e 1
8.b even 2 1 112.6.a.c 1
8.d odd 2 1 14.6.a.a 1
24.f even 2 1 126.6.a.f 1
24.h odd 2 1 1008.6.a.b 1
40.e odd 2 1 350.6.a.i 1
40.k even 4 2 350.6.c.d 2
56.e even 2 1 98.6.a.a 1
56.h odd 2 1 784.6.a.i 1
56.k odd 6 2 98.6.c.c 2
56.m even 6 2 98.6.c.d 2
168.e odd 2 1 882.6.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 8.d odd 2 1
98.6.a.a 1 56.e even 2 1
98.6.c.c 2 56.k odd 6 2
98.6.c.d 2 56.m even 6 2
112.6.a.c 1 8.b even 2 1
126.6.a.f 1 24.f even 2 1
350.6.a.i 1 40.e odd 2 1
350.6.c.d 2 40.k even 4 2
448.6.a.e 1 4.b odd 2 1
448.6.a.l 1 1.a even 1 1 trivial
784.6.a.i 1 56.h odd 2 1
882.6.a.x 1 168.e odd 2 1
1008.6.a.b 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3} - 10 \) Copy content Toggle raw display
\( T_{5} + 84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 10 \) Copy content Toggle raw display
$5$ \( T + 84 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T + 336 \) Copy content Toggle raw display
$13$ \( T + 584 \) Copy content Toggle raw display
$17$ \( T + 1458 \) Copy content Toggle raw display
$19$ \( T - 470 \) Copy content Toggle raw display
$23$ \( T - 4200 \) Copy content Toggle raw display
$29$ \( T + 4866 \) Copy content Toggle raw display
$31$ \( T - 7372 \) Copy content Toggle raw display
$37$ \( T + 14330 \) Copy content Toggle raw display
$41$ \( T - 6222 \) Copy content Toggle raw display
$43$ \( T - 3704 \) Copy content Toggle raw display
$47$ \( T - 1812 \) Copy content Toggle raw display
$53$ \( T - 37242 \) Copy content Toggle raw display
$59$ \( T - 34302 \) Copy content Toggle raw display
$61$ \( T + 24476 \) Copy content Toggle raw display
$67$ \( T + 17452 \) Copy content Toggle raw display
$71$ \( T + 28224 \) Copy content Toggle raw display
$73$ \( T - 3602 \) Copy content Toggle raw display
$79$ \( T + 42872 \) Copy content Toggle raw display
$83$ \( T + 35202 \) Copy content Toggle raw display
$89$ \( T - 26730 \) Copy content Toggle raw display
$97$ \( T + 16978 \) Copy content Toggle raw display
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