Properties

Label 448.4.a.e
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 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} - 16q^{5} + 7q^{7} - 23q^{9} + O(q^{10}) \) \( q - 2q^{3} - 16q^{5} + 7q^{7} - 23q^{9} - 8q^{11} - 28q^{13} + 32q^{15} + 54q^{17} - 110q^{19} - 14q^{21} - 48q^{23} + 131q^{25} + 100q^{27} + 110q^{29} - 12q^{31} + 16q^{33} - 112q^{35} + 246q^{37} + 56q^{39} + 182q^{41} + 128q^{43} + 368q^{45} - 324q^{47} + 49q^{49} - 108q^{51} + 162q^{53} + 128q^{55} + 220q^{57} + 810q^{59} + 488q^{61} - 161q^{63} + 448q^{65} + 244q^{67} + 96q^{69} + 768q^{71} - 702q^{73} - 262q^{75} - 56q^{77} - 440q^{79} + 421q^{81} - 1302q^{83} - 864q^{85} - 220q^{87} + 730q^{89} - 196q^{91} + 24q^{93} + 1760q^{95} + 294q^{97} + 184q^{99} + O(q^{100}) \)

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 −2.00000 0 −16.0000 0 7.00000 0 −23.0000 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.e 1
4.b odd 2 1 448.4.a.i 1
8.b even 2 1 112.4.a.f 1
8.d odd 2 1 7.4.a.a 1
24.f even 2 1 63.4.a.b 1
24.h odd 2 1 1008.4.a.c 1
40.e odd 2 1 175.4.a.b 1
40.k even 4 2 175.4.b.b 2
56.e even 2 1 49.4.a.b 1
56.h odd 2 1 784.4.a.g 1
56.k odd 6 2 49.4.c.c 2
56.m even 6 2 49.4.c.b 2
88.g even 2 1 847.4.a.b 1
104.h odd 2 1 1183.4.a.b 1
120.m even 2 1 1575.4.a.e 1
136.e odd 2 1 2023.4.a.a 1
168.e odd 2 1 441.4.a.i 1
168.v even 6 2 441.4.e.h 2
168.be odd 6 2 441.4.e.e 2
280.n even 2 1 1225.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 8.d odd 2 1
49.4.a.b 1 56.e even 2 1
49.4.c.b 2 56.m even 6 2
49.4.c.c 2 56.k odd 6 2
63.4.a.b 1 24.f even 2 1
112.4.a.f 1 8.b even 2 1
175.4.a.b 1 40.e odd 2 1
175.4.b.b 2 40.k even 4 2
441.4.a.i 1 168.e odd 2 1
441.4.e.e 2 168.be odd 6 2
441.4.e.h 2 168.v even 6 2
448.4.a.e 1 1.a even 1 1 trivial
448.4.a.i 1 4.b odd 2 1
784.4.a.g 1 56.h odd 2 1
847.4.a.b 1 88.g even 2 1
1008.4.a.c 1 24.h odd 2 1
1183.4.a.b 1 104.h odd 2 1
1225.4.a.j 1 280.n even 2 1
1575.4.a.e 1 120.m even 2 1
2023.4.a.a 1 136.e odd 2 1

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{5} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( 16 + T \)
$7$ \( -7 + T \)
$11$ \( 8 + T \)
$13$ \( 28 + T \)
$17$ \( -54 + T \)
$19$ \( 110 + T \)
$23$ \( 48 + T \)
$29$ \( -110 + T \)
$31$ \( 12 + T \)
$37$ \( -246 + T \)
$41$ \( -182 + T \)
$43$ \( -128 + T \)
$47$ \( 324 + T \)
$53$ \( -162 + T \)
$59$ \( -810 + T \)
$61$ \( -488 + T \)
$67$ \( -244 + T \)
$71$ \( -768 + T \)
$73$ \( 702 + T \)
$79$ \( 440 + T \)
$83$ \( 1302 + T \)
$89$ \( -730 + T \)
$97$ \( -294 + T \)
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