Properties

Label 224.2.i.a
Level $224$
Weight $2$
Character orbit 224.i
Analytic conductor $1.789$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{5} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{5} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} -2 \beta_{3} q^{13} + ( 5 + 3 \beta_{3} ) q^{15} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{17} + ( \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{19} + ( 3 - \beta_{2} + 2 \beta_{3} ) q^{21} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{23} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{25} + ( 1 - \beta_{3} ) q^{27} + 2 \beta_{3} q^{29} + ( -7 + \beta_{1} - 7 \beta_{2} ) q^{31} + \beta_{2} q^{33} + ( -8 + 3 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{35} + ( 4 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{37} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{39} + ( -4 + 2 \beta_{3} ) q^{41} + ( 4 - 4 \beta_{3} ) q^{43} + ( -8 + 2 \beta_{1} - 8 \beta_{2} ) q^{45} + ( \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{47} + ( -5 + 4 \beta_{1} + 2 \beta_{3} ) q^{49} + ( 5 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} ) q^{51} + ( 1 + \beta_{2} ) q^{53} + ( 3 - \beta_{3} ) q^{55} + ( 3 + 4 \beta_{3} ) q^{57} + ( -1 - 7 \beta_{1} - \beta_{2} ) q^{59} + ( 4 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{61} + ( -8 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{63} + ( 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 7 - 3 \beta_{1} + 7 \beta_{2} ) q^{67} + ( 5 + 2 \beta_{3} ) q^{69} + ( 8 - 4 \beta_{3} ) q^{71} + ( 9 + 4 \beta_{1} + 9 \beta_{2} ) q^{73} + ( -8 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} ) q^{75} + ( 1 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{77} + ( 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{79} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{81} + ( 4 + 8 \beta_{3} ) q^{83} + ( -11 - 8 \beta_{3} ) q^{85} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{87} -9 \beta_{2} q^{89} + ( -4 - 8 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -8 \beta_{1} + 9 \beta_{2} - 8 \beta_{3} ) q^{93} + ( -1 + 9 \beta_{1} - \beta_{2} ) q^{95} + ( -4 - 2 \beta_{3} ) q^{97} + ( 4 - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} - 4 q^{7} + O(q^{10}) \) \( 4 q - 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{11} + 20 q^{15} + 6 q^{17} - 10 q^{19} + 14 q^{21} + 2 q^{23} - 8 q^{25} + 4 q^{27} - 14 q^{31} - 2 q^{33} - 22 q^{35} - 6 q^{37} + 8 q^{39} - 16 q^{41} + 16 q^{43} - 16 q^{45} - 18 q^{47} - 20 q^{49} + 14 q^{51} + 2 q^{53} + 12 q^{55} + 12 q^{57} - 2 q^{59} - 6 q^{61} - 24 q^{63} + 16 q^{65} + 14 q^{67} + 20 q^{69} + 32 q^{71} + 18 q^{73} - 24 q^{75} + 10 q^{77} + 2 q^{79} + 2 q^{81} + 16 q^{83} - 44 q^{85} - 8 q^{87} + 18 q^{89} - 18 q^{93} - 2 q^{95} - 16 q^{97} + 16 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −1.20711 2.09077i 0 −1.91421 + 3.31552i 0 −1.00000 + 2.44949i 0 −1.41421 + 2.44949i 0
65.2 0 0.207107 + 0.358719i 0 0.914214 1.58346i 0 −1.00000 2.44949i 0 1.41421 2.44949i 0
193.1 0 −1.20711 + 2.09077i 0 −1.91421 3.31552i 0 −1.00000 2.44949i 0 −1.41421 2.44949i 0
193.2 0 0.207107 0.358719i 0 0.914214 + 1.58346i 0 −1.00000 + 2.44949i 0 1.41421 + 2.44949i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.i.a 4
3.b odd 2 1 2016.2.s.q 4
4.b odd 2 1 224.2.i.d yes 4
7.b odd 2 1 1568.2.i.x 4
7.c even 3 1 inner 224.2.i.a 4
7.c even 3 1 1568.2.a.w 2
7.d odd 6 1 1568.2.a.j 2
7.d odd 6 1 1568.2.i.x 4
8.b even 2 1 448.2.i.j 4
8.d odd 2 1 448.2.i.g 4
12.b even 2 1 2016.2.s.s 4
21.h odd 6 1 2016.2.s.q 4
28.d even 2 1 1568.2.i.o 4
28.f even 6 1 1568.2.a.u 2
28.f even 6 1 1568.2.i.o 4
28.g odd 6 1 224.2.i.d yes 4
28.g odd 6 1 1568.2.a.l 2
56.j odd 6 1 3136.2.a.bx 2
56.k odd 6 1 448.2.i.g 4
56.k odd 6 1 3136.2.a.bw 2
56.m even 6 1 3136.2.a.be 2
56.p even 6 1 448.2.i.j 4
56.p even 6 1 3136.2.a.bd 2
84.n even 6 1 2016.2.s.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a 4 1.a even 1 1 trivial
224.2.i.a 4 7.c even 3 1 inner
224.2.i.d yes 4 4.b odd 2 1
224.2.i.d yes 4 28.g odd 6 1
448.2.i.g 4 8.d odd 2 1
448.2.i.g 4 56.k odd 6 1
448.2.i.j 4 8.b even 2 1
448.2.i.j 4 56.p even 6 1
1568.2.a.j 2 7.d odd 6 1
1568.2.a.l 2 28.g odd 6 1
1568.2.a.u 2 28.f even 6 1
1568.2.a.w 2 7.c even 3 1
1568.2.i.o 4 28.d even 2 1
1568.2.i.o 4 28.f even 6 1
1568.2.i.x 4 7.b odd 2 1
1568.2.i.x 4 7.d odd 6 1
2016.2.s.q 4 3.b odd 2 1
2016.2.s.q 4 21.h odd 6 1
2016.2.s.s 4 12.b even 2 1
2016.2.s.s 4 84.n even 6 1
3136.2.a.bd 2 56.p even 6 1
3136.2.a.be 2 56.m even 6 1
3136.2.a.bw 2 56.k odd 6 1
3136.2.a.bx 2 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 49 - 14 T + 11 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( ( 7 + 2 T + T^{2} )^{2} \)
$11$ \( 1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( ( -8 + T^{2} )^{2} \)
$17$ \( 1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( 529 + 230 T + 77 T^{2} + 10 T^{3} + T^{4} \)
$23$ \( 289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( ( -8 + T^{2} )^{2} \)
$31$ \( 2209 + 658 T + 149 T^{2} + 14 T^{3} + T^{4} \)
$37$ \( 529 - 138 T + 59 T^{2} + 6 T^{3} + T^{4} \)
$41$ \( ( 8 + 8 T + T^{2} )^{2} \)
$43$ \( ( -16 - 8 T + T^{2} )^{2} \)
$47$ \( 6241 + 1422 T + 245 T^{2} + 18 T^{3} + T^{4} \)
$53$ \( ( 1 - T + T^{2} )^{2} \)
$59$ \( 9409 - 194 T + 101 T^{2} + 2 T^{3} + T^{4} \)
$61$ \( 529 - 138 T + 59 T^{2} + 6 T^{3} + T^{4} \)
$67$ \( 961 - 434 T + 165 T^{2} - 14 T^{3} + T^{4} \)
$71$ \( ( 32 - 16 T + T^{2} )^{2} \)
$73$ \( 2401 - 882 T + 275 T^{2} - 18 T^{3} + T^{4} \)
$79$ \( 2401 + 98 T + 53 T^{2} - 2 T^{3} + T^{4} \)
$83$ \( ( -112 - 8 T + T^{2} )^{2} \)
$89$ \( ( 81 - 9 T + T^{2} )^{2} \)
$97$ \( ( 8 + 8 T + T^{2} )^{2} \)
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