Properties

Label 224.3.h.b
Level 224
Weight 3
Character orbit 224.h
Self dual yes
Analytic conductor 6.104
Analytic rank 0
Dimension 2
CM discriminant -56
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: yes
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = 2\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{5} -7 q^{7} + 19 q^{9} +O(q^{10})\) \( q + \beta q^{3} + \beta q^{5} -7 q^{7} + 19 q^{9} + \beta q^{13} + 28 q^{15} -7 \beta q^{19} -7 \beta q^{21} + 10 q^{23} + 3 q^{25} + 10 \beta q^{27} -7 \beta q^{35} + 28 q^{39} + 19 \beta q^{45} + 49 q^{49} -196 q^{57} + 17 \beta q^{59} -23 \beta q^{61} -133 q^{63} + 28 q^{65} + 10 \beta q^{69} -110 q^{71} + 3 \beta q^{75} + 130 q^{79} + 109 q^{81} -31 \beta q^{83} -7 \beta q^{91} -196 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{7} + 38q^{9} + O(q^{10}) \) \( 2q - 14q^{7} + 38q^{9} + 56q^{15} + 20q^{23} + 6q^{25} + 56q^{39} + 98q^{49} - 392q^{57} - 266q^{63} + 56q^{65} - 220q^{71} + 260q^{79} + 218q^{81} - 392q^{95} + O(q^{100}) \)

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
−2.64575
2.64575
0 −5.29150 0 −5.29150 0 −7.00000 0 19.0000 0
209.2 0 5.29150 0 5.29150 0 −7.00000 0 19.0000 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
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
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 224.3.h.b 2
3.b odd 2 1 2016.3.l.a 2
4.b odd 2 1 56.3.h.a 2
7.b odd 2 1 inner 224.3.h.b 2
8.b even 2 1 inner 224.3.h.b 2
8.d odd 2 1 56.3.h.a 2
12.b even 2 1 504.3.l.c 2
21.c even 2 1 2016.3.l.a 2
24.f even 2 1 504.3.l.c 2
24.h odd 2 1 2016.3.l.a 2
28.d even 2 1 56.3.h.a 2
28.f even 6 2 392.3.j.c 4
28.g odd 6 2 392.3.j.c 4
56.e even 2 1 56.3.h.a 2
56.h odd 2 1 CM 224.3.h.b 2
56.k odd 6 2 392.3.j.c 4
56.m even 6 2 392.3.j.c 4
84.h odd 2 1 504.3.l.c 2
168.e odd 2 1 504.3.l.c 2
168.i even 2 1 2016.3.l.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.a 2 4.b odd 2 1
56.3.h.a 2 8.d odd 2 1
56.3.h.a 2 28.d even 2 1
56.3.h.a 2 56.e even 2 1
224.3.h.b 2 1.a even 1 1 trivial
224.3.h.b 2 7.b odd 2 1 inner
224.3.h.b 2 8.b even 2 1 inner
224.3.h.b 2 56.h odd 2 1 CM
392.3.j.c 4 28.f even 6 2
392.3.j.c 4 28.g odd 6 2
392.3.j.c 4 56.k odd 6 2
392.3.j.c 4 56.m even 6 2
504.3.l.c 2 12.b even 2 1
504.3.l.c 2 24.f even 2 1
504.3.l.c 2 84.h odd 2 1
504.3.l.c 2 168.e odd 2 1
2016.3.l.a 2 3.b odd 2 1
2016.3.l.a 2 21.c even 2 1
2016.3.l.a 2 24.h odd 2 1
2016.3.l.a 2 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 28 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 10 T^{2} + 81 T^{4} \)
$5$ \( 1 + 22 T^{2} + 625 T^{4} \)
$7$ \( ( 1 + 7 T )^{2} \)
$11$ \( ( 1 - 11 T )^{2}( 1 + 11 T )^{2} \)
$13$ \( 1 + 310 T^{2} + 28561 T^{4} \)
$17$ \( ( 1 - 17 T )^{2}( 1 + 17 T )^{2} \)
$19$ \( 1 - 650 T^{2} + 130321 T^{4} \)
$23$ \( ( 1 - 10 T + 529 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T )^{2}( 1 + 29 T )^{2} \)
$31$ \( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \)
$37$ \( ( 1 - 37 T )^{2}( 1 + 37 T )^{2} \)
$41$ \( ( 1 - 41 T )^{2}( 1 + 41 T )^{2} \)
$43$ \( ( 1 - 43 T )^{2}( 1 + 43 T )^{2} \)
$47$ \( ( 1 - 47 T )^{2}( 1 + 47 T )^{2} \)
$53$ \( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \)
$59$ \( 1 - 1130 T^{2} + 12117361 T^{4} \)
$61$ \( 1 - 7370 T^{2} + 13845841 T^{4} \)
$67$ \( ( 1 - 67 T )^{2}( 1 + 67 T )^{2} \)
$71$ \( ( 1 + 110 T + 5041 T^{2} )^{2} \)
$73$ \( ( 1 - 73 T )^{2}( 1 + 73 T )^{2} \)
$79$ \( ( 1 - 130 T + 6241 T^{2} )^{2} \)
$83$ \( 1 - 13130 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \)
$97$ \( ( 1 - 97 T )^{2}( 1 + 97 T )^{2} \)
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