Properties

Label 224.3.h.a
Level 224
Weight 3
Character orbit 224.h
Analytic conductor 6.104
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: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 224.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(\beta = 8\sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -7 q^{7} -9 q^{9} +O(q^{10})\) \( q -7 q^{7} -9 q^{9} -\beta q^{11} -18 q^{23} -25 q^{25} -\beta q^{29} -3 \beta q^{37} + 3 \beta q^{43} + 49 q^{49} + 5 \beta q^{53} + 63 q^{63} -3 \beta q^{67} + 114 q^{71} + 7 \beta q^{77} -94 q^{79} + 81 q^{81} + 9 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{7} - 18q^{9} + O(q^{10}) \) \( 2q - 14q^{7} - 18q^{9} - 36q^{23} - 50q^{25} + 98q^{49} + 126q^{63} + 228q^{71} - 188q^{79} + 162q^{81} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.500000 + 1.32288i
0.500000 1.32288i
0 0 0 0 0 −7.00000 0 −9.00000 0
209.2 0 0 0 0 0 −7.00000 0 −9.00000 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.b even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.h.a 2
3.b odd 2 1 2016.3.l.b 2
4.b odd 2 1 56.3.h.b 2
7.b odd 2 1 CM 224.3.h.a 2
8.b even 2 1 inner 224.3.h.a 2
8.d odd 2 1 56.3.h.b 2
12.b even 2 1 504.3.l.b 2
21.c even 2 1 2016.3.l.b 2
24.f even 2 1 504.3.l.b 2
24.h odd 2 1 2016.3.l.b 2
28.d even 2 1 56.3.h.b 2
28.f even 6 2 392.3.j.b 4
28.g odd 6 2 392.3.j.b 4
56.e even 2 1 56.3.h.b 2
56.h odd 2 1 inner 224.3.h.a 2
56.k odd 6 2 392.3.j.b 4
56.m even 6 2 392.3.j.b 4
84.h odd 2 1 504.3.l.b 2
168.e odd 2 1 504.3.l.b 2
168.i even 2 1 2016.3.l.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.b 2 4.b odd 2 1
56.3.h.b 2 8.d odd 2 1
56.3.h.b 2 28.d even 2 1
56.3.h.b 2 56.e even 2 1
224.3.h.a 2 1.a even 1 1 trivial
224.3.h.a 2 7.b odd 2 1 CM
224.3.h.a 2 8.b even 2 1 inner
224.3.h.a 2 56.h odd 2 1 inner
392.3.j.b 4 28.f even 6 2
392.3.j.b 4 28.g odd 6 2
392.3.j.b 4 56.k odd 6 2
392.3.j.b 4 56.m even 6 2
504.3.l.b 2 12.b even 2 1
504.3.l.b 2 24.f even 2 1
504.3.l.b 2 84.h odd 2 1
504.3.l.b 2 168.e odd 2 1
2016.3.l.b 2 3.b odd 2 1
2016.3.l.b 2 21.c even 2 1
2016.3.l.b 2 24.h odd 2 1
2016.3.l.b 2 168.i even 2 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 9 T^{2} )^{2} \)
$5$ \( ( 1 + 25 T^{2} )^{2} \)
$7$ \( ( 1 + 7 T )^{2} \)
$11$ \( ( 1 - 6 T + 121 T^{2} )( 1 + 6 T + 121 T^{2} ) \)
$13$ \( ( 1 + 169 T^{2} )^{2} \)
$17$ \( ( 1 - 17 T )^{2}( 1 + 17 T )^{2} \)
$19$ \( ( 1 + 361 T^{2} )^{2} \)
$23$ \( ( 1 + 18 T + 529 T^{2} )^{2} \)
$29$ \( ( 1 - 54 T + 841 T^{2} )( 1 + 54 T + 841 T^{2} ) \)
$31$ \( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \)
$37$ \( ( 1 - 38 T + 1369 T^{2} )( 1 + 38 T + 1369 T^{2} ) \)
$41$ \( ( 1 - 41 T )^{2}( 1 + 41 T )^{2} \)
$43$ \( ( 1 - 58 T + 1849 T^{2} )( 1 + 58 T + 1849 T^{2} ) \)
$47$ \( ( 1 - 47 T )^{2}( 1 + 47 T )^{2} \)
$53$ \( ( 1 - 6 T + 2809 T^{2} )( 1 + 6 T + 2809 T^{2} ) \)
$59$ \( ( 1 + 3481 T^{2} )^{2} \)
$61$ \( ( 1 + 3721 T^{2} )^{2} \)
$67$ \( ( 1 - 118 T + 4489 T^{2} )( 1 + 118 T + 4489 T^{2} ) \)
$71$ \( ( 1 - 114 T + 5041 T^{2} )^{2} \)
$73$ \( ( 1 - 73 T )^{2}( 1 + 73 T )^{2} \)
$79$ \( ( 1 + 94 T + 6241 T^{2} )^{2} \)
$83$ \( ( 1 + 6889 T^{2} )^{2} \)
$89$ \( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \)
$97$ \( ( 1 - 97 T )^{2}( 1 + 97 T )^{2} \)
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