Properties

Label 224.3.v.a
Level 224
Weight 3
Character orbit 224.v
Analytic conductor 6.104
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: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 224.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: 8.0.157351936.1
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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{6} - \beta_{7} ) q^{2} + ( -\beta_{3} + 3 \beta_{5} ) q^{4} -7 \beta_{6} q^{7} + ( 5 \beta_{1} - 2 \beta_{4} ) q^{8} -9 \beta_{4} q^{9} +O(q^{10})\) \( q + ( 2 \beta_{6} - \beta_{7} ) q^{2} + ( -\beta_{3} + 3 \beta_{5} ) q^{4} -7 \beta_{6} q^{7} + ( 5 \beta_{1} - 2 \beta_{4} ) q^{8} -9 \beta_{4} q^{9} + ( 8 \beta_{1} - \beta_{3} + \beta_{4} + 8 \beta_{5} ) q^{11} + ( -7 \beta_{3} - 7 \beta_{5} ) q^{14} + ( 14 + 3 \beta_{2} ) q^{16} + ( 18 - 9 \beta_{2} ) q^{18} + ( 14 + 15 \beta_{1} + 9 \beta_{2} - 2 \beta_{4} ) q^{22} + ( -17 + 16 \beta_{2} + 17 \beta_{3} - 16 \beta_{5} ) q^{23} -25 \beta_{6} q^{25} + ( -21 \beta_{1} - 14 \beta_{4} ) q^{28} + ( 31 \beta_{3} - 8 \beta_{5} - 31 \beta_{6} + 8 \beta_{7} ) q^{29} + ( 34 \beta_{6} - 11 \beta_{7} ) q^{32} + ( 18 \beta_{6} - 27 \beta_{7} ) q^{36} + ( -24 \beta_{1} - 31 \beta_{3} - 31 \beta_{4} + 24 \beta_{5} ) q^{37} + ( -41 + 24 \beta_{2} - 17 \beta_{6} - 24 \beta_{7} ) q^{43} + ( 34 + 13 \beta_{2} + 46 \beta_{6} - 5 \beta_{7} ) q^{44} + ( -15 \beta_{1} + 34 \beta_{4} - 2 \beta_{6} + 33 \beta_{7} ) q^{46} + 49 \beta_{3} q^{49} + ( -25 \beta_{3} - 25 \beta_{5} ) q^{50} + ( 23 - 40 \beta_{2} - 17 \beta_{6} + 40 \beta_{7} ) q^{53} + ( -14 - 35 \beta_{2} ) q^{56} + ( 15 \beta_{1} - 7 \beta_{3} + 62 \beta_{4} - 39 \beta_{5} ) q^{58} -63 q^{63} + ( \beta_{3} + 45 \beta_{5} ) q^{64} + ( -71 - 24 \beta_{1} + 24 \beta_{2} + 47 \beta_{4} ) q^{67} + ( 64 \beta_{1} + 32 \beta_{4} ) q^{71} + ( -63 \beta_{3} + 45 \beta_{5} ) q^{72} + ( 14 + 17 \beta_{1} - 55 \beta_{2} - 62 \beta_{4} ) q^{74} + ( 7 - 56 \beta_{1} - 56 \beta_{2} - 49 \beta_{4} ) q^{77} + ( 48 \beta_{1} + 71 \beta_{4} + 23 \beta_{6} + 48 \beta_{7} ) q^{79} -81 \beta_{3} q^{81} + ( -89 \beta_{3} + 7 \beta_{5} - 34 \beta_{6} + 65 \beta_{7} ) q^{86} + ( 31 \beta_{3} + 51 \beta_{5} + 94 \beta_{6} - 21 \beta_{7} ) q^{88} + ( -98 + 19 \beta_{2} + 97 \beta_{3} - 35 \beta_{5} ) q^{92} + ( 49 \beta_{1} + 98 \beta_{4} ) q^{98} + ( -63 \beta_{3} + 72 \beta_{5} + 63 \beta_{6} - 72 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 124q^{16} + 108q^{18} + 148q^{22} - 72q^{23} - 232q^{43} + 324q^{44} + 24q^{53} - 252q^{56} - 504q^{63} - 472q^{67} - 108q^{74} - 168q^{77} - 708q^{92} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{6}\)

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
−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
−1.99607 + 0.125246i 0 3.96863 0.500000i 0 0 4.94975 4.94975i −7.85905 + 1.49509i −6.36396 6.36396i 0
13.2 −0.125246 + 1.99607i 0 −3.96863 0.500000i 0 0 4.94975 4.94975i 1.49509 7.85905i −6.36396 6.36396i 0
69.1 −1.99607 0.125246i 0 3.96863 + 0.500000i 0 0 4.94975 + 4.94975i −7.85905 1.49509i −6.36396 + 6.36396i 0
69.2 −0.125246 1.99607i 0 −3.96863 + 0.500000i 0 0 4.94975 + 4.94975i 1.49509 + 7.85905i −6.36396 + 6.36396i 0
125.1 0.125246 1.99607i 0 −3.96863 0.500000i 0 0 −4.94975 + 4.94975i −1.49509 + 7.85905i 6.36396 + 6.36396i 0
125.2 1.99607 0.125246i 0 3.96863 0.500000i 0 0 −4.94975 + 4.94975i 7.85905 1.49509i 6.36396 + 6.36396i 0
181.1 0.125246 + 1.99607i 0 −3.96863 + 0.500000i 0 0 −4.94975 4.94975i −1.49509 7.85905i 6.36396 6.36396i 0
181.2 1.99607 + 0.125246i 0 3.96863 + 0.500000i 0 0 −4.94975 4.94975i 7.85905 + 1.49509i 6.36396 6.36396i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.2
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.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 224.3.v.a 8
7.b odd 2 1 CM 224.3.v.a 8
32.g even 8 1 inner 224.3.v.a 8
224.v odd 8 1 inner 224.3.v.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.v.a 8 1.a even 1 1 trivial
224.3.v.a 8 7.b odd 2 1 CM
224.3.v.a 8 32.g even 8 1 inner
224.3.v.a 8 224.v odd 8 1 inner

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 31 T^{4} + 256 T^{8} \)
$3$ \( ( 1 + 6561 T^{8} )^{2} \)
$5$ \( ( 1 + 390625 T^{8} )^{2} \)
$7$ \( ( 1 + 2401 T^{4} )^{2} \)
$11$ \( ( 1 - 206 T^{2} + 14641 T^{4} )^{2}( 1 + 13154 T^{4} + 214358881 T^{8} ) \)
$13$ \( ( 1 + 815730721 T^{8} )^{2} \)
$17$ \( ( 1 + 289 T^{2} )^{8} \)
$19$ \( ( 1 + 16983563041 T^{8} )^{2} \)
$23$ \( ( 1 + 18 T + 529 T^{2} )^{4}( 1 - 734 T^{2} + 279841 T^{4} )^{2} \)
$29$ \( ( 1 + 1234 T^{2} + 707281 T^{4} )^{2}( 1 + 108194 T^{4} + 500246412961 T^{8} ) \)
$31$ \( ( 1 - 31 T )^{8}( 1 + 31 T )^{8} \)
$37$ \( ( 1 - 1294 T^{2} + 1874161 T^{4} )^{2}( 1 - 2073886 T^{4} + 3512479453921 T^{8} ) \)
$41$ \( ( 1 + 2825761 T^{4} )^{4} \)
$43$ \( ( 1 + 58 T + 1849 T^{2} )^{4}( 1 - 6726046 T^{4} + 11688200277601 T^{8} ) \)
$47$ \( ( 1 + 2209 T^{2} )^{8} \)
$53$ \( ( 1 - 6 T + 2809 T^{2} )^{4}( 1 + 15377762 T^{4} + 62259690411361 T^{8} ) \)
$59$ \( ( 1 + 146830437604321 T^{8} )^{2} \)
$61$ \( ( 1 + 191707312997281 T^{8} )^{2} \)
$67$ \( ( 1 + 118 T + 4489 T^{2} )^{4}( 1 - 15839326 T^{4} + 406067677556641 T^{8} ) \)
$71$ \( ( 1 - 42331966 T^{4} + 645753531245761 T^{8} )^{2} \)
$73$ \( ( 1 + 28398241 T^{4} )^{4} \)
$79$ \( ( 1 - 64606846 T^{4} + 1517108809906561 T^{8} )^{2} \)
$83$ \( ( 1 + 2252292232139041 T^{8} )^{2} \)
$89$ \( ( 1 + 62742241 T^{4} )^{4} \)
$97$ \( ( 1 - 97 T )^{8}( 1 + 97 T )^{8} \)
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