Properties

Label 224.2.u.a
Level $224$
Weight $2$
Character orbit 224.u
Analytic conductor $1.789$
Analytic rank $1$
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.u (of order \(8\), degree \(4\), minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{8}^{2} ) q^{2} + ( -1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{4} + ( -2 - 2 \zeta_{8} ) q^{5} + ( 2 + 2 \zeta_{8} ) q^{6} + \zeta_{8} q^{7} + ( 2 + 2 \zeta_{8}^{2} ) q^{8} + ( -2 + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{8}^{2} ) q^{2} + ( -1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{4} + ( -2 - 2 \zeta_{8} ) q^{5} + ( 2 + 2 \zeta_{8} ) q^{6} + \zeta_{8} q^{7} + ( 2 + 2 \zeta_{8}^{2} ) q^{8} + ( -2 + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{9} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{10} + ( -3 + 3 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{11} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{12} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{13} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{14} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{15} -4 q^{16} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{18} + ( -1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{19} + ( 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{20} + ( 1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{21} + ( 1 - 5 \zeta_{8} - 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{22} + ( -1 + \zeta_{8}^{2} ) q^{23} + ( -4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{24} + ( 4 + 3 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{25} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{26} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{27} -2 \zeta_{8}^{3} q^{28} + ( -4 - 3 \zeta_{8} - 3 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{29} + ( -4 - 8 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{30} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} + ( 4 - 4 \zeta_{8}^{2} ) q^{32} + ( 10 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{33} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{35} + ( 4 + 2 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{36} + ( -4 - 4 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{37} + ( 4 - 4 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{38} -4 \zeta_{8} q^{39} + ( -4 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{40} + ( -2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{42} + ( -4 + 4 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{43} + ( 4 + 4 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{44} + ( 6 + 4 \zeta_{8} - 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{45} -2 \zeta_{8}^{2} q^{46} + ( 2 \zeta_{8} + 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} + ( 4 + 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{48} + \zeta_{8}^{2} q^{49} + ( -8 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{50} + ( -4 - 4 \zeta_{8}^{3} ) q^{52} + ( 5 - 5 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{53} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{54} + ( 10 - 10 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{55} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{56} + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{57} + ( 7 + 7 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{58} + ( -5 - 5 \zeta_{8} + 5 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{59} + ( 8 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{60} + ( -4 - 4 \zeta_{8}^{3} ) q^{61} + ( 6 - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{62} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} + 8 \zeta_{8}^{2} q^{64} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{65} + ( -10 + 10 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{66} + ( 4 + 5 \zeta_{8} + 5 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{67} + ( 2 + 2 \zeta_{8} ) q^{69} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{70} -6 \zeta_{8} q^{71} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{72} + ( 6 - 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{73} + ( 1 + 7 \zeta_{8} - 7 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{74} + ( 3 - 3 \zeta_{8} - 11 \zeta_{8}^{2} - 11 \zeta_{8}^{3} ) q^{75} + ( -6 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{76} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{77} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{78} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{79} + ( 8 + 8 \zeta_{8} ) q^{80} + ( 2 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{82} + ( -1 + 7 \zeta_{8} - 7 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{83} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{84} + ( 3 - 5 \zeta_{8} - 5 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{86} + ( -6 + 6 \zeta_{8}^{2} + 14 \zeta_{8}^{3} ) q^{87} + ( -10 + 2 \zeta_{8} - 2 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{88} + ( -8 + 4 \zeta_{8} - 8 \zeta_{8}^{2} ) q^{89} + ( -2 + 2 \zeta_{8} + 10 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{90} + ( 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{91} + ( 2 + 2 \zeta_{8}^{2} ) q^{92} + ( 6 + 10 \zeta_{8} + 10 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{93} + ( -6 - 4 \zeta_{8} - 6 \zeta_{8}^{2} ) q^{94} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} + ( -8 - 8 \zeta_{8} ) q^{96} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{97} + ( -1 - \zeta_{8}^{2} ) q^{98} + ( -1 - 12 \zeta_{8} - 12 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} - 8 q^{5} + 8 q^{6} + 8 q^{8} - 8 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} - 4 q^{3} - 8 q^{5} + 8 q^{6} + 8 q^{8} - 8 q^{9} + 8 q^{10} - 12 q^{11} - 8 q^{12} - 16 q^{16} - 4 q^{19} + 4 q^{21} + 4 q^{22} - 4 q^{23} + 16 q^{25} + 8 q^{26} + 8 q^{27} - 16 q^{29} - 16 q^{30} - 24 q^{31} + 16 q^{32} + 40 q^{33} + 16 q^{36} - 16 q^{37} + 16 q^{38} - 16 q^{40} - 8 q^{41} - 16 q^{43} + 16 q^{44} + 24 q^{45} + 16 q^{48} - 32 q^{50} - 16 q^{52} + 20 q^{53} + 40 q^{55} + 8 q^{57} + 28 q^{58} - 20 q^{59} + 32 q^{60} - 16 q^{61} + 24 q^{62} - 4 q^{63} - 40 q^{66} + 16 q^{67} + 8 q^{69} + 8 q^{70} - 32 q^{72} + 24 q^{73} + 4 q^{74} + 12 q^{75} - 24 q^{76} - 8 q^{77} + 32 q^{80} - 4 q^{83} - 8 q^{84} + 12 q^{86} - 24 q^{87} - 40 q^{88} - 32 q^{89} - 8 q^{90} + 8 q^{92} + 24 q^{93} - 24 q^{94} + 16 q^{95} - 32 q^{96} + 16 q^{97} - 4 q^{98} - 4 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−1.00000 1.00000i −1.00000 0.414214i 2.00000i −0.585786 1.41421i 0.585786 + 1.41421i −0.707107 + 0.707107i 2.00000 2.00000i −1.29289 1.29289i −0.828427 + 2.00000i
85.1 −1.00000 + 1.00000i −1.00000 + 0.414214i 2.00000i −0.585786 + 1.41421i 0.585786 1.41421i −0.707107 0.707107i 2.00000 + 2.00000i −1.29289 + 1.29289i −0.828427 2.00000i
141.1 −1.00000 1.00000i −1.00000 + 2.41421i 2.00000i −3.41421 + 1.41421i 3.41421 1.41421i 0.707107 0.707107i 2.00000 2.00000i −2.70711 2.70711i 4.82843 + 2.00000i
197.1 −1.00000 + 1.00000i −1.00000 2.41421i 2.00000i −3.41421 1.41421i 3.41421 + 1.41421i 0.707107 + 0.707107i 2.00000 + 2.00000i −2.70711 + 2.70711i 4.82843 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g 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.u.a 4
4.b odd 2 1 896.2.u.a 4
32.g even 8 1 inner 224.2.u.a 4
32.h odd 8 1 896.2.u.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.a 4 1.a even 1 1 trivial
224.2.u.a 4 32.g even 8 1 inner
896.2.u.a 4 4.b odd 2 1
896.2.u.a 4 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4 T_{3}^{3} + 12 T_{3}^{2} + 16 T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + 2 T + T^{2} )^{2} \)
$3$ \( 8 + 16 T + 12 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( 32 + 32 T + 24 T^{2} + 8 T^{3} + T^{4} \)
$7$ \( 1 + T^{4} \)
$11$ \( 578 + 340 T + 86 T^{2} + 12 T^{3} + T^{4} \)
$13$ \( 32 - 32 T + 8 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( 8 - 32 T + 36 T^{2} + 4 T^{3} + T^{4} \)
$23$ \( ( 2 + 2 T + T^{2} )^{2} \)
$29$ \( 1922 + 868 T + 162 T^{2} + 16 T^{3} + T^{4} \)
$31$ \( ( 28 + 12 T + T^{2} )^{2} \)
$37$ \( 1922 + 868 T + 162 T^{2} + 16 T^{3} + T^{4} \)
$41$ \( 16 + 32 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( 1058 + 460 T + 114 T^{2} + 16 T^{3} + T^{4} \)
$47$ \( 784 + 88 T^{2} + T^{4} \)
$53$ \( 2 - 12 T + 118 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( 5000 + 2000 T + 300 T^{2} + 20 T^{3} + T^{4} \)
$61$ \( 512 + 256 T + 96 T^{2} + 16 T^{3} + T^{4} \)
$67$ \( 1922 - 1116 T + 226 T^{2} - 16 T^{3} + T^{4} \)
$71$ \( 1296 + T^{4} \)
$73$ \( 3136 - 1344 T + 288 T^{2} - 24 T^{3} + T^{4} \)
$79$ \( ( 98 + T^{2} )^{2} \)
$83$ \( 2312 - 1088 T + 132 T^{2} + 4 T^{3} + T^{4} \)
$89$ \( 12544 + 3584 T + 512 T^{2} + 32 T^{3} + T^{4} \)
$97$ \( ( 8 - 8 T + T^{2} )^{2} \)
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