Properties

Label 224.2.b.a
Level $224$
Weight $2$
Character orbit 224.b
Analytic conductor $1.789$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{5} - q^{7} + q^{9} + 2 \beta q^{11} + 3 \beta q^{13} - 2 q^{15} - 6 q^{17} - 3 \beta q^{19} - \beta q^{21} + 6 q^{23} + 3 q^{25} + 4 \beta q^{27} - 2 \beta q^{29} + 4 q^{31} - 4 q^{33} - \beta q^{35} - 6 \beta q^{37} - 6 q^{39} + 6 q^{41} - 6 \beta q^{43} + \beta q^{45} + q^{49} - 6 \beta q^{51} + 4 \beta q^{53} - 4 q^{55} + 6 q^{57} - \beta q^{59} - 9 \beta q^{61} - q^{63} - 6 q^{65} + 6 \beta q^{69} + 2 q^{73} + 3 \beta q^{75} - 2 \beta q^{77} - 8 q^{79} - 5 q^{81} + 11 \beta q^{83} - 6 \beta q^{85} + 4 q^{87} + 6 q^{89} - 3 \beta q^{91} + 4 \beta q^{93} + 6 q^{95} - 10 q^{97} + 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} + 2 q^{9} - 4 q^{15} - 12 q^{17} + 12 q^{23} + 6 q^{25} + 8 q^{31} - 8 q^{33} - 12 q^{39} + 12 q^{41} + 2 q^{49} - 8 q^{55} + 12 q^{57} - 2 q^{63} - 12 q^{65} + 4 q^{73} - 16 q^{79} - 10 q^{81} + 8 q^{87} + 12 q^{89} + 12 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
1.41421i
1.41421i
0 1.41421i 0 1.41421i 0 −1.00000 0 1.00000 0
113.2 0 1.41421i 0 1.41421i 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.b.a 2
3.b odd 2 1 2016.2.c.a 2
4.b odd 2 1 56.2.b.a 2
7.b odd 2 1 1568.2.b.a 2
7.c even 3 2 1568.2.t.c 4
7.d odd 6 2 1568.2.t.b 4
8.b even 2 1 inner 224.2.b.a 2
8.d odd 2 1 56.2.b.a 2
12.b even 2 1 504.2.c.a 2
16.e even 4 2 1792.2.a.p 2
16.f odd 4 2 1792.2.a.n 2
24.f even 2 1 504.2.c.a 2
24.h odd 2 1 2016.2.c.a 2
28.d even 2 1 392.2.b.b 2
28.f even 6 2 392.2.p.b 4
28.g odd 6 2 392.2.p.a 4
56.e even 2 1 392.2.b.b 2
56.h odd 2 1 1568.2.b.a 2
56.j odd 6 2 1568.2.t.b 4
56.k odd 6 2 392.2.p.a 4
56.m even 6 2 392.2.p.b 4
56.p even 6 2 1568.2.t.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.a 2 4.b odd 2 1
56.2.b.a 2 8.d odd 2 1
224.2.b.a 2 1.a even 1 1 trivial
224.2.b.a 2 8.b even 2 1 inner
392.2.b.b 2 28.d even 2 1
392.2.b.b 2 56.e even 2 1
392.2.p.a 4 28.g odd 6 2
392.2.p.a 4 56.k odd 6 2
392.2.p.b 4 28.f even 6 2
392.2.p.b 4 56.m even 6 2
504.2.c.a 2 12.b even 2 1
504.2.c.a 2 24.f even 2 1
1568.2.b.a 2 7.b odd 2 1
1568.2.b.a 2 56.h odd 2 1
1568.2.t.b 4 7.d odd 6 2
1568.2.t.b 4 56.j odd 6 2
1568.2.t.c 4 7.c even 3 2
1568.2.t.c 4 56.p even 6 2
1792.2.a.n 2 16.f odd 4 2
1792.2.a.p 2 16.e even 4 2
2016.2.c.a 2 3.b odd 2 1
2016.2.c.a 2 24.h odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 18 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 18 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 8 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 72 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 72 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 32 \) Copy content Toggle raw display
$59$ \( T^{2} + 2 \) Copy content Toggle raw display
$61$ \( T^{2} + 162 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 242 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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