Properties

Label 224.1.r.a
Level 224
Weight 1
Character orbit 224.r
Analytic conductor 0.112
Analytic rank 0
Dimension 4
Projective image \(A_{4}\)
CM/RM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 224.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.111790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.3136.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{3} q^{7} -\zeta_{12}^{5} q^{11} -\zeta_{12}^{3} q^{15} -\zeta_{12}^{2} q^{17} + \zeta_{12} q^{19} -\zeta_{12}^{2} q^{21} -\zeta_{12} q^{23} -\zeta_{12}^{3} q^{27} -\zeta_{12}^{5} q^{31} + \zeta_{12}^{4} q^{33} -\zeta_{12} q^{35} -\zeta_{12}^{4} q^{37} + \zeta_{12} q^{47} - q^{49} + \zeta_{12} q^{51} + \zeta_{12}^{2} q^{53} + \zeta_{12}^{3} q^{55} - q^{57} + \zeta_{12}^{5} q^{59} -\zeta_{12}^{4} q^{61} + \zeta_{12}^{5} q^{67} + q^{69} + \zeta_{12}^{2} q^{73} + \zeta_{12}^{2} q^{77} -\zeta_{12} q^{79} + \zeta_{12}^{2} q^{81} + q^{85} -\zeta_{12}^{4} q^{89} + \zeta_{12}^{4} q^{93} + \zeta_{12}^{5} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + O(q^{10}) \) \( 4q - 2q^{5} - 2q^{17} - 2q^{21} - 2q^{33} + 2q^{37} - 4q^{49} + 2q^{53} - 4q^{57} + 2q^{61} + 4q^{69} + 2q^{73} + 2q^{77} + 2q^{81} + 4q^{85} + 2q^{89} - 2q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 0.500000i 0 −0.500000 0.866025i 0 1.00000i 0 0 0
95.2 0 0.866025 + 0.500000i 0 −0.500000 0.866025i 0 1.00000i 0 0 0
191.1 0 −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 1.00000i 0 0 0
191.2 0 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 1.00000i 0 0 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.1.r.a 4
3.b odd 2 1 2016.1.cd.a 4
4.b odd 2 1 inner 224.1.r.a 4
7.b odd 2 1 1568.1.r.a 4
7.c even 3 1 inner 224.1.r.a 4
7.c even 3 1 1568.1.d.b 2
7.d odd 6 1 1568.1.d.a 2
7.d odd 6 1 1568.1.r.a 4
8.b even 2 1 448.1.r.a 4
8.d odd 2 1 448.1.r.a 4
12.b even 2 1 2016.1.cd.a 4
16.e even 4 1 1792.1.o.a 4
16.e even 4 1 1792.1.o.b 4
16.f odd 4 1 1792.1.o.a 4
16.f odd 4 1 1792.1.o.b 4
21.h odd 6 1 2016.1.cd.a 4
28.d even 2 1 1568.1.r.a 4
28.f even 6 1 1568.1.d.a 2
28.f even 6 1 1568.1.r.a 4
28.g odd 6 1 inner 224.1.r.a 4
28.g odd 6 1 1568.1.d.b 2
56.e even 2 1 3136.1.r.b 4
56.h odd 2 1 3136.1.r.b 4
56.j odd 6 1 3136.1.d.d 2
56.j odd 6 1 3136.1.r.b 4
56.k odd 6 1 448.1.r.a 4
56.k odd 6 1 3136.1.d.b 2
56.m even 6 1 3136.1.d.d 2
56.m even 6 1 3136.1.r.b 4
56.p even 6 1 448.1.r.a 4
56.p even 6 1 3136.1.d.b 2
84.n even 6 1 2016.1.cd.a 4
112.u odd 12 1 1792.1.o.a 4
112.u odd 12 1 1792.1.o.b 4
112.w even 12 1 1792.1.o.a 4
112.w even 12 1 1792.1.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 1.a even 1 1 trivial
224.1.r.a 4 4.b odd 2 1 inner
224.1.r.a 4 7.c even 3 1 inner
224.1.r.a 4 28.g odd 6 1 inner
448.1.r.a 4 8.b even 2 1
448.1.r.a 4 8.d odd 2 1
448.1.r.a 4 56.k odd 6 1
448.1.r.a 4 56.p even 6 1
1568.1.d.a 2 7.d odd 6 1
1568.1.d.a 2 28.f even 6 1
1568.1.d.b 2 7.c even 3 1
1568.1.d.b 2 28.g odd 6 1
1568.1.r.a 4 7.b odd 2 1
1568.1.r.a 4 7.d odd 6 1
1568.1.r.a 4 28.d even 2 1
1568.1.r.a 4 28.f even 6 1
1792.1.o.a 4 16.e even 4 1
1792.1.o.a 4 16.f odd 4 1
1792.1.o.a 4 112.u odd 12 1
1792.1.o.a 4 112.w even 12 1
1792.1.o.b 4 16.e even 4 1
1792.1.o.b 4 16.f odd 4 1
1792.1.o.b 4 112.u odd 12 1
1792.1.o.b 4 112.w even 12 1
2016.1.cd.a 4 3.b odd 2 1
2016.1.cd.a 4 12.b even 2 1
2016.1.cd.a 4 21.h odd 6 1
2016.1.cd.a 4 84.n even 6 1
3136.1.d.b 2 56.k odd 6 1
3136.1.d.b 2 56.p even 6 1
3136.1.d.d 2 56.j odd 6 1
3136.1.d.d 2 56.m even 6 1
3136.1.r.b 4 56.e even 2 1
3136.1.r.b 4 56.h odd 2 1
3136.1.r.b 4 56.j odd 6 1
3136.1.r.b 4 56.m even 6 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
$3$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$5$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$13$ \( ( 1 + T^{2} )^{4} \)
$17$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$19$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$23$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$29$ \( ( 1 + T^{2} )^{4} \)
$31$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$37$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$41$ \( ( 1 + T^{2} )^{4} \)
$43$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$47$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$53$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$59$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$61$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$67$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$71$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$73$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$79$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$83$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$89$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$97$ \( ( 1 + T^{2} )^{4} \)
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