Properties

Label 448.4.a.a
Level $448$
Weight $4$
Character orbit 448.a
Self dual yes
Analytic conductor $26.433$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.4328556826\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - 10q^{3} + 8q^{5} + 7q^{7} + 73q^{9} + O(q^{10}) \) \( q - 10q^{3} + 8q^{5} + 7q^{7} + 73q^{9} - 40q^{11} + 12q^{13} - 80q^{15} - 58q^{17} + 26q^{19} - 70q^{21} + 64q^{23} - 61q^{25} - 460q^{27} + 62q^{29} - 252q^{31} + 400q^{33} + 56q^{35} - 26q^{37} - 120q^{39} + 6q^{41} + 416q^{43} + 584q^{45} + 396q^{47} + 49q^{49} + 580q^{51} + 450q^{53} - 320q^{55} - 260q^{57} + 274q^{59} + 576q^{61} + 511q^{63} + 96q^{65} - 476q^{67} - 640q^{69} + 448q^{71} - 158q^{73} + 610q^{75} - 280q^{77} + 936q^{79} + 2629q^{81} + 530q^{83} - 464q^{85} - 620q^{87} - 390q^{89} + 84q^{91} + 2520q^{93} + 208q^{95} + 214q^{97} - 2920q^{99} + O(q^{100}) \)

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 8.00000 0 7.00000 0 73.0000 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 448.4.a.a 1
4.b odd 2 1 448.4.a.p 1
8.b even 2 1 112.4.a.g 1
8.d odd 2 1 28.4.a.a 1
24.f even 2 1 252.4.a.d 1
24.h odd 2 1 1008.4.a.o 1
40.e odd 2 1 700.4.a.n 1
40.k even 4 2 700.4.e.a 2
56.e even 2 1 196.4.a.d 1
56.h odd 2 1 784.4.a.a 1
56.k odd 6 2 196.4.e.f 2
56.m even 6 2 196.4.e.a 2
168.e odd 2 1 1764.4.a.c 1
168.v even 6 2 1764.4.k.d 2
168.be odd 6 2 1764.4.k.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 8.d odd 2 1
112.4.a.g 1 8.b even 2 1
196.4.a.d 1 56.e even 2 1
196.4.e.a 2 56.m even 6 2
196.4.e.f 2 56.k odd 6 2
252.4.a.d 1 24.f even 2 1
448.4.a.a 1 1.a even 1 1 trivial
448.4.a.p 1 4.b odd 2 1
700.4.a.n 1 40.e odd 2 1
700.4.e.a 2 40.k even 4 2
784.4.a.a 1 56.h odd 2 1
1008.4.a.o 1 24.h odd 2 1
1764.4.a.c 1 168.e odd 2 1
1764.4.k.d 2 168.v even 6 2
1764.4.k.m 2 168.be odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3} + 10 \)
\( T_{5} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 10 + T \)
$5$ \( -8 + T \)
$7$ \( -7 + T \)
$11$ \( 40 + T \)
$13$ \( -12 + T \)
$17$ \( 58 + T \)
$19$ \( -26 + T \)
$23$ \( -64 + T \)
$29$ \( -62 + T \)
$31$ \( 252 + T \)
$37$ \( 26 + T \)
$41$ \( -6 + T \)
$43$ \( -416 + T \)
$47$ \( -396 + T \)
$53$ \( -450 + T \)
$59$ \( -274 + T \)
$61$ \( -576 + T \)
$67$ \( 476 + T \)
$71$ \( -448 + T \)
$73$ \( 158 + T \)
$79$ \( -936 + T \)
$83$ \( -530 + T \)
$89$ \( 390 + T \)
$97$ \( -214 + T \)
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