Properties

Label 224.2.a.d
Level $224$
Weight $2$
Character orbit 224.a
Self dual yes
Analytic conductor $1.789$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + ( - \beta + 1) q^{5} - q^{7} + (2 \beta + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + ( - \beta + 1) q^{5} - q^{7} + (2 \beta + 3) q^{9} + ( - 2 \beta + 2) q^{11} + (\beta + 3) q^{13} - 4 q^{15} - 2 \beta q^{17} + ( - \beta - 1) q^{19} + ( - \beta - 1) q^{21} - 4 q^{23} + ( - 2 \beta + 1) q^{25} + (2 \beta + 10) q^{27} + 2 \beta q^{29} + ( - 2 \beta - 2) q^{31} - 8 q^{33} + (\beta - 1) q^{35} + 2 \beta q^{37} + (4 \beta + 8) q^{39} + (2 \beta - 4) q^{41} + (2 \beta - 2) q^{43} + ( - \beta - 7) q^{45} + (2 \beta - 6) q^{47} + q^{49} + ( - 2 \beta - 10) q^{51} - 10 q^{53} + ( - 4 \beta + 12) q^{55} + ( - 2 \beta - 6) q^{57} + ( - \beta + 7) q^{59} + ( - \beta + 9) q^{61} + ( - 2 \beta - 3) q^{63} + ( - 2 \beta - 2) q^{65} - 4 q^{67} + ( - 4 \beta - 4) q^{69} + ( - 4 \beta - 4) q^{71} + (4 \beta + 6) q^{73} + ( - \beta - 9) q^{75} + (2 \beta - 2) q^{77} + (4 \beta - 4) q^{79} + (6 \beta + 11) q^{81} + ( - \beta + 7) q^{83} + ( - 2 \beta + 10) q^{85} + (2 \beta + 10) q^{87} - 6 q^{89} + ( - \beta - 3) q^{91} + ( - 4 \beta - 12) q^{93} + 4 q^{95} + ( - 2 \beta + 8) q^{97} + ( - 2 \beta - 14) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{31} - 16 q^{33} - 2 q^{35} + 16 q^{39} - 8 q^{41} - 4 q^{43} - 14 q^{45} - 12 q^{47} + 2 q^{49} - 20 q^{51} - 20 q^{53} + 24 q^{55} - 12 q^{57} + 14 q^{59} + 18 q^{61} - 6 q^{63} - 4 q^{65} - 8 q^{67} - 8 q^{69} - 8 q^{71} + 12 q^{73} - 18 q^{75} - 4 q^{77} - 8 q^{79} + 22 q^{81} + 14 q^{83} + 20 q^{85} + 20 q^{87} - 12 q^{89} - 6 q^{91} - 24 q^{93} + 8 q^{95} + 16 q^{97} - 28 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
0 −1.23607 0 3.23607 0 −1.00000 0 −1.47214 0
1.2 0 3.23607 0 −1.23607 0 −1.00000 0 7.47214 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 224.2.a.d yes 2
3.b odd 2 1 2016.2.a.o 2
4.b odd 2 1 224.2.a.c 2
5.b even 2 1 5600.2.a.z 2
7.b odd 2 1 1568.2.a.k 2
7.c even 3 2 1568.2.i.m 4
7.d odd 6 2 1568.2.i.w 4
8.b even 2 1 448.2.a.i 2
8.d odd 2 1 448.2.a.j 2
12.b even 2 1 2016.2.a.r 2
16.e even 4 2 1792.2.b.m 4
16.f odd 4 2 1792.2.b.k 4
20.d odd 2 1 5600.2.a.bk 2
24.f even 2 1 4032.2.a.bw 2
24.h odd 2 1 4032.2.a.bv 2
28.d even 2 1 1568.2.a.v 2
28.f even 6 2 1568.2.i.n 4
28.g odd 6 2 1568.2.i.v 4
56.e even 2 1 3136.2.a.bf 2
56.h odd 2 1 3136.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 4.b odd 2 1
224.2.a.d yes 2 1.a even 1 1 trivial
448.2.a.i 2 8.b even 2 1
448.2.a.j 2 8.d odd 2 1
1568.2.a.k 2 7.b odd 2 1
1568.2.a.v 2 28.d even 2 1
1568.2.i.m 4 7.c even 3 2
1568.2.i.n 4 28.f even 6 2
1568.2.i.v 4 28.g odd 6 2
1568.2.i.w 4 7.d odd 6 2
1792.2.b.k 4 16.f odd 4 2
1792.2.b.m 4 16.e even 4 2
2016.2.a.o 2 3.b odd 2 1
2016.2.a.r 2 12.b even 2 1
3136.2.a.bf 2 56.e even 2 1
3136.2.a.by 2 56.h odd 2 1
4032.2.a.bv 2 24.h odd 2 1
4032.2.a.bw 2 24.f even 2 1
5600.2.a.z 2 5.b even 2 1
5600.2.a.bk 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(224))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 20 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
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