Properties

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

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{3} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + \beta_{3} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -3 q^{9} -2 q^{11} -\beta_{3} q^{13} + 6 \beta_{1} q^{15} + 2 \beta_{2} q^{17} -\beta_{2} q^{19} + ( 6 \beta_{1} + \beta_{3} ) q^{21} -4 \beta_{1} q^{23} + q^{25} -4 \beta_{1} q^{29} -2 \beta_{3} q^{31} -2 \beta_{2} q^{33} + ( 6 - \beta_{2} ) q^{35} -8 \beta_{1} q^{37} -6 \beta_{1} q^{39} + 6 q^{43} -3 \beta_{3} q^{45} + 2 \beta_{3} q^{47} + ( 5 - 2 \beta_{2} ) q^{49} -12 q^{51} + 4 \beta_{1} q^{53} -2 \beta_{3} q^{55} + 6 q^{57} -\beta_{2} q^{59} -3 \beta_{3} q^{61} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{63} -6 q^{65} + 2 q^{67} + 4 \beta_{3} q^{69} + 10 \beta_{1} q^{71} -6 \beta_{2} q^{73} + \beta_{2} q^{75} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{77} -6 \beta_{1} q^{79} -9 q^{81} + \beta_{2} q^{83} + 12 \beta_{1} q^{85} + 4 \beta_{3} q^{87} + 6 \beta_{2} q^{89} + ( -6 + \beta_{2} ) q^{91} -12 \beta_{1} q^{93} -6 \beta_{1} q^{95} + 2 \beta_{2} q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} - 8q^{11} + 4q^{25} + 24q^{35} + 24q^{43} + 20q^{49} - 48q^{51} + 24q^{57} - 24q^{65} + 8q^{67} - 36q^{81} - 24q^{91} + 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
−1.22474 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
0 2.44949i 0 −2.44949 0 −2.44949 1.00000i 0 −3.00000 0
111.2 0 2.44949i 0 2.44949 0 2.44949 + 1.00000i 0 −3.00000 0
111.3 0 2.44949i 0 −2.44949 0 −2.44949 + 1.00000i 0 −3.00000 0
111.4 0 2.44949i 0 2.44949 0 2.44949 1.00000i 0 −3.00000 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 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.e.b 4
3.b odd 2 1 2016.2.p.e 4
4.b odd 2 1 56.2.e.b 4
7.b odd 2 1 inner 224.2.e.b 4
7.c even 3 2 1568.2.q.e 8
7.d odd 6 2 1568.2.q.e 8
8.b even 2 1 56.2.e.b 4
8.d odd 2 1 inner 224.2.e.b 4
12.b even 2 1 504.2.p.f 4
16.e even 4 1 1792.2.f.e 4
16.e even 4 1 1792.2.f.f 4
16.f odd 4 1 1792.2.f.e 4
16.f odd 4 1 1792.2.f.f 4
21.c even 2 1 2016.2.p.e 4
24.f even 2 1 2016.2.p.e 4
24.h odd 2 1 504.2.p.f 4
28.d even 2 1 56.2.e.b 4
28.f even 6 2 392.2.m.f 8
28.g odd 6 2 392.2.m.f 8
56.e even 2 1 inner 224.2.e.b 4
56.h odd 2 1 56.2.e.b 4
56.j odd 6 2 392.2.m.f 8
56.k odd 6 2 1568.2.q.e 8
56.m even 6 2 1568.2.q.e 8
56.p even 6 2 392.2.m.f 8
84.h odd 2 1 504.2.p.f 4
112.j even 4 1 1792.2.f.e 4
112.j even 4 1 1792.2.f.f 4
112.l odd 4 1 1792.2.f.e 4
112.l odd 4 1 1792.2.f.f 4
168.e odd 2 1 2016.2.p.e 4
168.i even 2 1 504.2.p.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.b 4 4.b odd 2 1
56.2.e.b 4 8.b even 2 1
56.2.e.b 4 28.d even 2 1
56.2.e.b 4 56.h odd 2 1
224.2.e.b 4 1.a even 1 1 trivial
224.2.e.b 4 7.b odd 2 1 inner
224.2.e.b 4 8.d odd 2 1 inner
224.2.e.b 4 56.e even 2 1 inner
392.2.m.f 8 28.f even 6 2
392.2.m.f 8 28.g odd 6 2
392.2.m.f 8 56.j odd 6 2
392.2.m.f 8 56.p even 6 2
504.2.p.f 4 12.b even 2 1
504.2.p.f 4 24.h odd 2 1
504.2.p.f 4 84.h odd 2 1
504.2.p.f 4 168.i even 2 1
1568.2.q.e 8 7.c even 3 2
1568.2.q.e 8 7.d odd 6 2
1568.2.q.e 8 56.k odd 6 2
1568.2.q.e 8 56.m even 6 2
1792.2.f.e 4 16.e even 4 1
1792.2.f.e 4 16.f odd 4 1
1792.2.f.e 4 112.j even 4 1
1792.2.f.e 4 112.l odd 4 1
1792.2.f.f 4 16.e even 4 1
1792.2.f.f 4 16.f odd 4 1
1792.2.f.f 4 112.j even 4 1
1792.2.f.f 4 112.l odd 4 1
2016.2.p.e 4 3.b odd 2 1
2016.2.p.e 4 21.c even 2 1
2016.2.p.e 4 24.f even 2 1
2016.2.p.e 4 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 6 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + 9 T^{4} )^{2} \)
$5$ \( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \)
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{4} \)
$13$ \( ( 1 + 20 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 38 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 - 6 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 112 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 68 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 2 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 42 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 70 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 38 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \)
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