Properties

Label 448.2.a.e
Level $448$
Weight $2$
Character orbit 448.a
Self dual yes
Analytic conductor $3.577$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(3.57729801055\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{5} + q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} + q^{7} - 3 q^{9} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 8 q^{19} - q^{25} - 6 q^{29} - 8 q^{31} - 2 q^{35} + 2 q^{37} + 2 q^{41} - 4 q^{43} + 6 q^{45} + 8 q^{47} + q^{49} - 6 q^{53} + 8 q^{55} + 6 q^{61} - 3 q^{63} + 4 q^{65} - 4 q^{67} + 8 q^{71} + 10 q^{73} - 4 q^{77} - 16 q^{79} + 9 q^{81} + 8 q^{83} + 12 q^{85} - 6 q^{89} - 2 q^{91} - 16 q^{95} - 6 q^{97} + 12 q^{99}+O(q^{100}) \) 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 0 0 −2.00000 0 1.00000 0 −3.00000 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 448.2.a.e 1
3.b odd 2 1 4032.2.a.bk 1
4.b odd 2 1 448.2.a.d 1
7.b odd 2 1 3136.2.a.p 1
8.b even 2 1 112.2.a.b 1
8.d odd 2 1 56.2.a.a 1
12.b even 2 1 4032.2.a.bb 1
16.e even 4 2 1792.2.b.d 2
16.f odd 4 2 1792.2.b.i 2
24.f even 2 1 504.2.a.c 1
24.h odd 2 1 1008.2.a.d 1
28.d even 2 1 3136.2.a.q 1
40.e odd 2 1 1400.2.a.g 1
40.f even 2 1 2800.2.a.p 1
40.i odd 4 2 2800.2.g.p 2
40.k even 4 2 1400.2.g.g 2
56.e even 2 1 392.2.a.d 1
56.h odd 2 1 784.2.a.e 1
56.j odd 6 2 784.2.i.g 2
56.k odd 6 2 392.2.i.c 2
56.m even 6 2 392.2.i.d 2
56.p even 6 2 784.2.i.e 2
88.g even 2 1 6776.2.a.g 1
104.h odd 2 1 9464.2.a.c 1
168.e odd 2 1 3528.2.a.x 1
168.i even 2 1 7056.2.a.bo 1
168.v even 6 2 3528.2.s.t 2
168.be odd 6 2 3528.2.s.e 2
280.n even 2 1 9800.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 8.d odd 2 1
112.2.a.b 1 8.b even 2 1
392.2.a.d 1 56.e even 2 1
392.2.i.c 2 56.k odd 6 2
392.2.i.d 2 56.m even 6 2
448.2.a.d 1 4.b odd 2 1
448.2.a.e 1 1.a even 1 1 trivial
504.2.a.c 1 24.f even 2 1
784.2.a.e 1 56.h odd 2 1
784.2.i.e 2 56.p even 6 2
784.2.i.g 2 56.j odd 6 2
1008.2.a.d 1 24.h odd 2 1
1400.2.a.g 1 40.e odd 2 1
1400.2.g.g 2 40.k even 4 2
1792.2.b.d 2 16.e even 4 2
1792.2.b.i 2 16.f odd 4 2
2800.2.a.p 1 40.f even 2 1
2800.2.g.p 2 40.i odd 4 2
3136.2.a.p 1 7.b odd 2 1
3136.2.a.q 1 28.d even 2 1
3528.2.a.x 1 168.e odd 2 1
3528.2.s.e 2 168.be odd 6 2
3528.2.s.t 2 168.v even 6 2
4032.2.a.bb 1 12.b even 2 1
4032.2.a.bk 1 3.b odd 2 1
6776.2.a.g 1 88.g even 2 1
7056.2.a.bo 1 168.i even 2 1
9464.2.a.c 1 104.h odd 2 1
9800.2.a.u 1 280.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T - 8 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 6 \) Copy content Toggle raw display
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