Properties

Label 224.2.a.a
Level 224
Weight 2
Character orbit 224.a
Self dual yes
Analytic conductor 1.789
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.78864900528\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} - q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{3} - q^{7} + q^{9} - 4q^{11} - 4q^{13} - 2q^{17} - 6q^{19} + 2q^{21} + 8q^{23} - 5q^{25} + 4q^{27} + 2q^{29} - 4q^{31} + 8q^{33} + 10q^{37} + 8q^{39} - 10q^{41} + 4q^{43} + 4q^{47} + q^{49} + 4q^{51} - 2q^{53} + 12q^{57} + 10q^{59} - 8q^{61} - q^{63} - 8q^{67} - 16q^{69} - 6q^{73} + 10q^{75} + 4q^{77} - 16q^{79} - 11q^{81} + 2q^{83} - 4q^{87} + 18q^{89} + 4q^{91} + 8q^{93} - 2q^{97} - 4q^{99} + O(q^{100}) \)

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 0 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 224.2.a.a 1
3.b odd 2 1 2016.2.a.e 1
4.b odd 2 1 224.2.a.b yes 1
5.b even 2 1 5600.2.a.t 1
7.b odd 2 1 1568.2.a.h 1
7.c even 3 2 1568.2.i.k 2
7.d odd 6 2 1568.2.i.c 2
8.b even 2 1 448.2.a.f 1
8.d odd 2 1 448.2.a.b 1
12.b even 2 1 2016.2.a.g 1
16.e even 4 2 1792.2.b.f 2
16.f odd 4 2 1792.2.b.b 2
20.d odd 2 1 5600.2.a.c 1
24.f even 2 1 4032.2.a.z 1
24.h odd 2 1 4032.2.a.p 1
28.d even 2 1 1568.2.a.b 1
28.f even 6 2 1568.2.i.j 2
28.g odd 6 2 1568.2.i.b 2
56.e even 2 1 3136.2.a.y 1
56.h odd 2 1 3136.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 1.a even 1 1 trivial
224.2.a.b yes 1 4.b odd 2 1
448.2.a.b 1 8.d odd 2 1
448.2.a.f 1 8.b even 2 1
1568.2.a.b 1 28.d even 2 1
1568.2.a.h 1 7.b odd 2 1
1568.2.i.b 2 28.g odd 6 2
1568.2.i.c 2 7.d odd 6 2
1568.2.i.j 2 28.f even 6 2
1568.2.i.k 2 7.c even 3 2
1792.2.b.b 2 16.f odd 4 2
1792.2.b.f 2 16.e even 4 2
2016.2.a.e 1 3.b odd 2 1
2016.2.a.g 1 12.b even 2 1
3136.2.a.f 1 56.h odd 2 1
3136.2.a.y 1 56.e even 2 1
4032.2.a.p 1 24.h odd 2 1
4032.2.a.z 1 24.f even 2 1
5600.2.a.c 1 20.d odd 2 1
5600.2.a.t 1 5.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(224))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 2 T + 3 T^{2} \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 - 8 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 - 10 T + 37 T^{2} \)
$41$ \( 1 + 10 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 - 4 T + 47 T^{2} \)
$53$ \( 1 + 2 T + 53 T^{2} \)
$59$ \( 1 - 10 T + 59 T^{2} \)
$61$ \( 1 + 8 T + 61 T^{2} \)
$67$ \( 1 + 8 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 + 16 T + 79 T^{2} \)
$83$ \( 1 - 2 T + 83 T^{2} \)
$89$ \( 1 - 18 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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