Properties

Label 448.6.a.o
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 28)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 26 q^{3} - 16 q^{5} + 49 q^{7} + 433 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 26 q^{3} - 16 q^{5} + 49 q^{7} + 433 q^{9} + 8 q^{11} - 684 q^{13} - 416 q^{15} - 2218 q^{17} - 2698 q^{19} + 1274 q^{21} - 3344 q^{23} - 2869 q^{25} + 4940 q^{27} + 3254 q^{29} - 4788 q^{31} + 208 q^{33} - 784 q^{35} + 11470 q^{37} - 17784 q^{39} + 13350 q^{41} - 928 q^{43} - 6928 q^{45} - 1212 q^{47} + 2401 q^{49} - 57668 q^{51} - 13110 q^{53} - 128 q^{55} - 70148 q^{57} + 34702 q^{59} + 1032 q^{61} + 21217 q^{63} + 10944 q^{65} + 10108 q^{67} - 86944 q^{69} - 62720 q^{71} - 18926 q^{73} - 74594 q^{75} + 392 q^{77} - 11400 q^{79} + 23221 q^{81} + 88958 q^{83} + 35488 q^{85} + 84604 q^{87} + 19722 q^{89} - 33516 q^{91} - 124488 q^{93} + 43168 q^{95} + 17062 q^{97} + 3464 q^{99}+O(q^{100}) \) 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 26.0000 0 −16.0000 0 49.0000 0 433.000 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.o 1
4.b odd 2 1 448.6.a.b 1
8.b even 2 1 112.6.a.b 1
8.d odd 2 1 28.6.a.b 1
24.f even 2 1 252.6.a.a 1
24.h odd 2 1 1008.6.a.l 1
40.e odd 2 1 700.6.a.b 1
40.k even 4 2 700.6.e.b 2
56.e even 2 1 196.6.a.a 1
56.h odd 2 1 784.6.a.m 1
56.k odd 6 2 196.6.e.a 2
56.m even 6 2 196.6.e.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.b 1 8.d odd 2 1
112.6.a.b 1 8.b even 2 1
196.6.a.a 1 56.e even 2 1
196.6.e.a 2 56.k odd 6 2
196.6.e.i 2 56.m even 6 2
252.6.a.a 1 24.f even 2 1
448.6.a.b 1 4.b odd 2 1
448.6.a.o 1 1.a even 1 1 trivial
700.6.a.b 1 40.e odd 2 1
700.6.e.b 2 40.k even 4 2
784.6.a.m 1 56.h odd 2 1
1008.6.a.l 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3} - 26 \) Copy content Toggle raw display
\( T_{5} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 26 \) Copy content Toggle raw display
$5$ \( T + 16 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 8 \) Copy content Toggle raw display
$13$ \( T + 684 \) Copy content Toggle raw display
$17$ \( T + 2218 \) Copy content Toggle raw display
$19$ \( T + 2698 \) Copy content Toggle raw display
$23$ \( T + 3344 \) Copy content Toggle raw display
$29$ \( T - 3254 \) Copy content Toggle raw display
$31$ \( T + 4788 \) Copy content Toggle raw display
$37$ \( T - 11470 \) Copy content Toggle raw display
$41$ \( T - 13350 \) Copy content Toggle raw display
$43$ \( T + 928 \) Copy content Toggle raw display
$47$ \( T + 1212 \) Copy content Toggle raw display
$53$ \( T + 13110 \) Copy content Toggle raw display
$59$ \( T - 34702 \) Copy content Toggle raw display
$61$ \( T - 1032 \) Copy content Toggle raw display
$67$ \( T - 10108 \) Copy content Toggle raw display
$71$ \( T + 62720 \) Copy content Toggle raw display
$73$ \( T + 18926 \) Copy content Toggle raw display
$79$ \( T + 11400 \) Copy content Toggle raw display
$83$ \( T - 88958 \) Copy content Toggle raw display
$89$ \( T - 19722 \) Copy content Toggle raw display
$97$ \( T - 17062 \) Copy content Toggle raw display
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