Properties

Label 224.2.x.a
Level 224
Weight 2
Character orbit 224.x
Analytic conductor 1.789
Analytic rank 0
Dimension 8
CM discriminant -7
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{4} ) q^{2} + ( \beta_{3} + \beta_{5} ) q^{4} + ( 2 \beta_{2} + \beta_{4} ) q^{7} + ( 2 \beta_{6} + \beta_{7} ) q^{8} -3 \beta_{6} q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{4} ) q^{2} + ( \beta_{3} + \beta_{5} ) q^{4} + ( 2 \beta_{2} + \beta_{4} ) q^{7} + ( 2 \beta_{6} + \beta_{7} ) q^{8} -3 \beta_{6} q^{9} + ( -3 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{11} + ( 3 \beta_{3} + \beta_{5} ) q^{14} + ( -2 + 3 \beta_{1} ) q^{16} -3 \beta_{1} q^{18} + ( -4 - \beta_{1} + 4 \beta_{6} - 3 \beta_{7} ) q^{22} + ( 5 - 2 \beta_{1} - 5 \beta_{3} + 2 \beta_{5} ) q^{23} -5 \beta_{4} q^{25} + ( 2 \beta_{6} + 3 \beta_{7} ) q^{28} + ( -4 \beta_{2} + \beta_{3} - 3 \beta_{4} - 4 \beta_{5} ) q^{29} + ( \beta_{2} - 5 \beta_{4} ) q^{32} + ( -3 \beta_{2} + 3 \beta_{4} ) q^{36} + ( -\beta_{3} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{37} + ( -7 + 2 \beta_{1} + 2 \beta_{2} + 7 \beta_{4} ) q^{43} + ( 6 + \beta_{1} - 5 \beta_{2} - 3 \beta_{4} ) q^{44} + ( 3 \beta_{2} + 7 \beta_{4} + 4 \beta_{6} - 5 \beta_{7} ) q^{46} + 7 \beta_{3} q^{49} + ( 5 \beta_{3} - 5 \beta_{5} ) q^{50} + ( 3 + 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{4} ) q^{53} + ( -6 + 5 \beta_{1} ) q^{56} + ( -5 \beta_{3} - 3 \beta_{5} - 8 \beta_{6} + \beta_{7} ) q^{58} + ( 3 - 6 \beta_{1} ) q^{63} + ( 7 \beta_{3} - 5 \beta_{5} ) q^{64} + ( -1 + 6 \beta_{1} - 5 \beta_{6} + 6 \beta_{7} ) q^{67} + 16 \beta_{6} q^{71} + ( -9 \beta_{3} + 3 \beta_{5} ) q^{72} + ( 8 - 5 \beta_{1} - 8 \beta_{6} - \beta_{7} ) q^{74} + ( -5 - 4 \beta_{1} + 9 \beta_{6} - 4 \beta_{7} ) q^{77} + ( 6 \beta_{2} - \beta_{4} - \beta_{6} - 6 \beta_{7} ) q^{79} + 9 \beta_{3} q^{81} + ( -5 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} + 7 \beta_{5} ) q^{86} + ( 7 \beta_{2} - 7 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} ) q^{88} + ( 10 - \beta_{1} - \beta_{3} + 7 \beta_{5} ) q^{92} + 7 \beta_{7} q^{98} + ( -6 \beta_{2} + 3 \beta_{3} - 9 \beta_{4} + 6 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 4q^{16} - 12q^{18} - 36q^{22} + 32q^{23} - 48q^{43} + 52q^{44} + 40q^{53} - 28q^{56} + 16q^{67} + 44q^{74} - 56q^{77} + 76q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} + 2 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 5 \nu \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + \nu \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 7 \nu^{2} \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 7 \nu^{3} \)\()/24\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{3} \)\()/12\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{4} + \beta_{2}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{6}\)
\(\nu^{4}\)\(=\)\(3 \beta_{1} - 2\)
\(\nu^{5}\)\(=\)\(-5 \beta_{4} + \beta_{2}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{5} + 7 \beta_{3}\)
\(\nu^{7}\)\(=\)\(7 \beta_{7} - 10 \beta_{6}\)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
−1.28897 + 0.581861i
0.581861 1.28897i
−1.28897 0.581861i
0.581861 + 1.28897i
−0.581861 + 1.28897i
1.28897 0.581861i
−0.581861 1.28897i
1.28897 + 0.581861i
−1.28897 + 0.581861i 0 1.32288 1.50000i 0 0 −1.87083 + 1.87083i −0.832353 + 2.70318i 2.12132 2.12132i 0
27.2 0.581861 1.28897i 0 −1.32288 1.50000i 0 0 1.87083 1.87083i −2.70318 + 0.832353i 2.12132 2.12132i 0
83.1 −1.28897 0.581861i 0 1.32288 + 1.50000i 0 0 −1.87083 1.87083i −0.832353 2.70318i 2.12132 + 2.12132i 0
83.2 0.581861 + 1.28897i 0 −1.32288 + 1.50000i 0 0 1.87083 + 1.87083i −2.70318 0.832353i 2.12132 + 2.12132i 0
139.1 −0.581861 + 1.28897i 0 −1.32288 1.50000i 0 0 −1.87083 + 1.87083i 2.70318 0.832353i −2.12132 + 2.12132i 0
139.2 1.28897 0.581861i 0 1.32288 1.50000i 0 0 1.87083 1.87083i 0.832353 2.70318i −2.12132 + 2.12132i 0
195.1 −0.581861 1.28897i 0 −1.32288 + 1.50000i 0 0 −1.87083 1.87083i 2.70318 + 0.832353i −2.12132 2.12132i 0
195.2 1.28897 + 0.581861i 0 1.32288 + 1.50000i 0 0 1.87083 + 1.87083i 0.832353 + 2.70318i −2.12132 2.12132i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 195.2
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.h odd 8 1 inner
224.x even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.x.a 8
4.b odd 2 1 896.2.x.a 8
7.b odd 2 1 CM 224.2.x.a 8
28.d even 2 1 896.2.x.a 8
32.g even 8 1 896.2.x.a 8
32.h odd 8 1 inner 224.2.x.a 8
224.v odd 8 1 896.2.x.a 8
224.x even 8 1 inner 224.2.x.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.x.a 8 1.a even 1 1 trivial
224.2.x.a 8 7.b odd 2 1 CM
224.2.x.a 8 32.h odd 8 1 inner
224.2.x.a 8 224.x even 8 1 inner
896.2.x.a 8 4.b odd 2 1
896.2.x.a 8 28.d even 2 1
896.2.x.a 8 32.g even 8 1
896.2.x.a 8 224.v odd 8 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} + 16 T^{8} \)
$3$ \( ( 1 + 81 T^{8} )^{2} \)
$5$ \( ( 1 + 625 T^{8} )^{2} \)
$7$ \( ( 1 + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 6 T^{2} + 121 T^{4} )^{2}( 1 - 206 T^{4} + 14641 T^{8} ) \)
$13$ \( ( 1 + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{8} \)
$19$ \( ( 1 + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 8 T + 23 T^{2} )^{4}( 1 + 18 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 54 T^{2} + 841 T^{4} )^{2}( 1 + 1234 T^{4} + 707281 T^{8} ) \)
$31$ \( ( 1 + 31 T^{2} )^{8} \)
$37$ \( ( 1 - 38 T^{2} + 1369 T^{4} )^{2}( 1 - 1294 T^{4} + 1874161 T^{8} ) \)
$41$ \( ( 1 + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 12 T + 43 T^{2} )^{4}( 1 - 334 T^{4} + 3418801 T^{8} ) \)
$47$ \( ( 1 - 47 T^{2} )^{8} \)
$53$ \( ( 1 - 10 T + 53 T^{2} )^{4}( 1 - 5582 T^{4} + 7890481 T^{8} ) \)
$59$ \( ( 1 + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4946 T^{4} + 20151121 T^{8} ) \)
$71$ \( ( 1 + 2914 T^{4} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 3646 T^{4} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 97 T^{2} )^{8} \)
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