Properties

Label 224.1.v.a
Level 224
Weight 1
Character orbit 224.v
Analytic conductor 0.112
Analytic rank 0
Dimension 4
Projective image \(D_{8}\)
CM discriminant -7
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 224.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.111790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{8}\)
Projective field Galois closure of 8.0.5156108238848.3

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + \zeta_{8}^{3} q^{9} +O(q^{10})\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + \zeta_{8}^{3} q^{9} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{11} -\zeta_{8}^{2} q^{14} - q^{16} - q^{18} + ( 1 - \zeta_{8}^{3} ) q^{22} + ( -1 + \zeta_{8}^{2} ) q^{23} + \zeta_{8} q^{25} -\zeta_{8}^{3} q^{28} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{29} -\zeta_{8} q^{32} -\zeta_{8} q^{36} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{37} + ( -1 + \zeta_{8} ) q^{43} + ( 1 + \zeta_{8} ) q^{44} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{46} + \zeta_{8}^{2} q^{49} + \zeta_{8}^{2} q^{50} + ( -1 + \zeta_{8} ) q^{53} + q^{56} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{58} + q^{63} -\zeta_{8}^{2} q^{64} + ( 1 + \zeta_{8}^{3} ) q^{67} -\zeta_{8}^{2} q^{72} + ( 1 + \zeta_{8}^{3} ) q^{74} + ( -1 + \zeta_{8}^{3} ) q^{77} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{79} -\zeta_{8}^{2} q^{81} + ( -\zeta_{8} + \zeta_{8}^{2} ) q^{86} + ( \zeta_{8} + \zeta_{8}^{2} ) q^{88} + ( -1 - \zeta_{8}^{2} ) q^{92} + \zeta_{8}^{3} q^{98} + ( \zeta_{8} + \zeta_{8}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{16} - 4q^{18} + 4q^{22} - 4q^{23} - 4q^{43} + 4q^{44} - 4q^{53} + 4q^{56} + 4q^{63} + 4q^{67} + 4q^{74} - 4q^{77} - 4q^{92} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 0.707107i −0.707107 0.707107i 0
69.1 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 + 0.707107i −0.707107 + 0.707107i 0
125.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 + 0.707107i 0.707107 + 0.707107i 0
181.1 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 0.707107i 0.707107 0.707107i 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 \(\Q(\sqrt{-7}) \)
32.g even 8 1 inner
224.v odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.1.v.a 4
3.b odd 2 1 2016.1.dp.b 4
4.b odd 2 1 896.1.v.a 4
7.b odd 2 1 CM 224.1.v.a 4
7.c even 3 2 1568.1.bl.a 8
7.d odd 6 2 1568.1.bl.a 8
8.b even 2 1 1792.1.v.a 4
8.d odd 2 1 1792.1.v.b 4
16.e even 4 1 3584.1.v.b 4
16.e even 4 1 3584.1.v.d 4
16.f odd 4 1 3584.1.v.a 4
16.f odd 4 1 3584.1.v.c 4
21.c even 2 1 2016.1.dp.b 4
28.d even 2 1 896.1.v.a 4
32.g even 8 1 inner 224.1.v.a 4
32.g even 8 1 1792.1.v.a 4
32.g even 8 1 3584.1.v.b 4
32.g even 8 1 3584.1.v.d 4
32.h odd 8 1 896.1.v.a 4
32.h odd 8 1 1792.1.v.b 4
32.h odd 8 1 3584.1.v.a 4
32.h odd 8 1 3584.1.v.c 4
56.e even 2 1 1792.1.v.b 4
56.h odd 2 1 1792.1.v.a 4
96.p odd 8 1 2016.1.dp.b 4
112.j even 4 1 3584.1.v.a 4
112.j even 4 1 3584.1.v.c 4
112.l odd 4 1 3584.1.v.b 4
112.l odd 4 1 3584.1.v.d 4
224.v odd 8 1 inner 224.1.v.a 4
224.v odd 8 1 1792.1.v.a 4
224.v odd 8 1 3584.1.v.b 4
224.v odd 8 1 3584.1.v.d 4
224.x even 8 1 896.1.v.a 4
224.x even 8 1 1792.1.v.b 4
224.x even 8 1 3584.1.v.a 4
224.x even 8 1 3584.1.v.c 4
224.bc odd 24 2 1568.1.bl.a 8
224.bd even 24 2 1568.1.bl.a 8
672.bo even 8 1 2016.1.dp.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.v.a 4 1.a even 1 1 trivial
224.1.v.a 4 7.b odd 2 1 CM
224.1.v.a 4 32.g even 8 1 inner
224.1.v.a 4 224.v odd 8 1 inner
896.1.v.a 4 4.b odd 2 1
896.1.v.a 4 28.d even 2 1
896.1.v.a 4 32.h odd 8 1
896.1.v.a 4 224.x even 8 1
1568.1.bl.a 8 7.c even 3 2
1568.1.bl.a 8 7.d odd 6 2
1568.1.bl.a 8 224.bc odd 24 2
1568.1.bl.a 8 224.bd even 24 2
1792.1.v.a 4 8.b even 2 1
1792.1.v.a 4 32.g even 8 1
1792.1.v.a 4 56.h odd 2 1
1792.1.v.a 4 224.v odd 8 1
1792.1.v.b 4 8.d odd 2 1
1792.1.v.b 4 32.h odd 8 1
1792.1.v.b 4 56.e even 2 1
1792.1.v.b 4 224.x even 8 1
2016.1.dp.b 4 3.b odd 2 1
2016.1.dp.b 4 21.c even 2 1
2016.1.dp.b 4 96.p odd 8 1
2016.1.dp.b 4 672.bo even 8 1
3584.1.v.a 4 16.f odd 4 1
3584.1.v.a 4 32.h odd 8 1
3584.1.v.a 4 112.j even 4 1
3584.1.v.a 4 224.x even 8 1
3584.1.v.b 4 16.e even 4 1
3584.1.v.b 4 32.g even 8 1
3584.1.v.b 4 112.l odd 4 1
3584.1.v.b 4 224.v odd 8 1
3584.1.v.c 4 16.f odd 4 1
3584.1.v.c 4 32.h odd 8 1
3584.1.v.c 4 112.j even 4 1
3584.1.v.c 4 224.x even 8 1
3584.1.v.d 4 16.e even 4 1
3584.1.v.d 4 32.g even 8 1
3584.1.v.d 4 112.l odd 4 1
3584.1.v.d 4 224.v odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

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