Properties

Label 224.2.e.a
Level 224
Weight 2
Character orbit 224.e
Analytic conductor 1.789
Analytic rank 0
Dimension 2
CM discriminant -7
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{7} + 3 q^{9} +O(q^{10})\) \( q -\beta q^{7} + 3 q^{9} + 4 q^{11} + 2 \beta q^{23} -5 q^{25} -4 \beta q^{29} + 4 \beta q^{37} -12 q^{43} -7 q^{49} + 4 \beta q^{53} -3 \beta q^{63} -4 q^{67} -2 \beta q^{71} -4 \beta q^{77} + 6 \beta q^{79} + 9 q^{81} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{9} + O(q^{10}) \) \( 2q + 6q^{9} + 8q^{11} - 10q^{25} - 24q^{43} - 14q^{49} - 8q^{67} + 18q^{81} + 24q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
0.500000 + 1.32288i
0.500000 1.32288i
0 0 0 0 0 2.64575i 0 3.00000 0
111.2 0 0 0 0 0 2.64575i 0 3.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.d odd 2 1 inner
56.e 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.e.a 2
3.b odd 2 1 2016.2.p.a 2
4.b odd 2 1 56.2.e.a 2
7.b odd 2 1 CM 224.2.e.a 2
7.c even 3 2 1568.2.q.a 4
7.d odd 6 2 1568.2.q.a 4
8.b even 2 1 56.2.e.a 2
8.d odd 2 1 inner 224.2.e.a 2
12.b even 2 1 504.2.p.a 2
16.e even 4 2 1792.2.f.d 4
16.f odd 4 2 1792.2.f.d 4
21.c even 2 1 2016.2.p.a 2
24.f even 2 1 2016.2.p.a 2
24.h odd 2 1 504.2.p.a 2
28.d even 2 1 56.2.e.a 2
28.f even 6 2 392.2.m.a 4
28.g odd 6 2 392.2.m.a 4
56.e even 2 1 inner 224.2.e.a 2
56.h odd 2 1 56.2.e.a 2
56.j odd 6 2 392.2.m.a 4
56.k odd 6 2 1568.2.q.a 4
56.m even 6 2 1568.2.q.a 4
56.p even 6 2 392.2.m.a 4
84.h odd 2 1 504.2.p.a 2
112.j even 4 2 1792.2.f.d 4
112.l odd 4 2 1792.2.f.d 4
168.e odd 2 1 2016.2.p.a 2
168.i even 2 1 504.2.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.a 2 4.b odd 2 1
56.2.e.a 2 8.b even 2 1
56.2.e.a 2 28.d even 2 1
56.2.e.a 2 56.h odd 2 1
224.2.e.a 2 1.a even 1 1 trivial
224.2.e.a 2 7.b odd 2 1 CM
224.2.e.a 2 8.d odd 2 1 inner
224.2.e.a 2 56.e even 2 1 inner
392.2.m.a 4 28.f even 6 2
392.2.m.a 4 28.g odd 6 2
392.2.m.a 4 56.j odd 6 2
392.2.m.a 4 56.p even 6 2
504.2.p.a 2 12.b even 2 1
504.2.p.a 2 24.h odd 2 1
504.2.p.a 2 84.h odd 2 1
504.2.p.a 2 168.i even 2 1
1568.2.q.a 4 7.c even 3 2
1568.2.q.a 4 7.d odd 6 2
1568.2.q.a 4 56.k odd 6 2
1568.2.q.a 4 56.m even 6 2
1792.2.f.d 4 16.e even 4 2
1792.2.f.d 4 16.f odd 4 2
1792.2.f.d 4 112.j even 4 2
1792.2.f.d 4 112.l odd 4 2
2016.2.p.a 2 3.b odd 2 1
2016.2.p.a 2 21.c even 2 1
2016.2.p.a 2 24.f even 2 1
2016.2.p.a 2 168.e odd 2 1

Hecke kernels

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

Hecke Characteristic Polynomials

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