Properties

Label 360.2.bd.a
Level $360$
Weight $2$
Character orbit 360.bd
Analytic conductor $2.875$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{2} + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{3} + 2 \zeta_{24}^{4} q^{4} + ( 2 \zeta_{24} + \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{6} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} -3 \zeta_{24}^{4} q^{9} +O(q^{10})\) \( q + ( -\zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{2} + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{3} + 2 \zeta_{24}^{4} q^{4} + ( 2 \zeta_{24} + \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{6} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} -3 \zeta_{24}^{4} q^{9} + ( -1 - 3 \zeta_{24}^{6} ) q^{10} + ( 3 + 3 \zeta_{24}^{4} ) q^{11} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{12} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{13} + ( 4 - 2 \zeta_{24}^{4} ) q^{14} + ( \zeta_{24} + 2 \zeta_{24}^{3} + \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{15} + ( -4 + 4 \zeta_{24}^{4} ) q^{16} + ( 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{17} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{18} -5 q^{19} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{20} + ( 3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{21} + ( 3 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{22} + ( -2 \zeta_{24} + 2 \zeta_{24}^{7} ) q^{23} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{24} + ( -4 + 3 \zeta_{24}^{2} + 4 \zeta_{24}^{4} ) q^{25} + ( 2 - 4 \zeta_{24}^{4} ) q^{26} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{27} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{28} + ( 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{29} + ( -3 - \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{30} + ( 3 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{31} + ( 4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{32} + ( 9 \zeta_{24}^{2} - 9 \zeta_{24}^{6} ) q^{33} + ( -6 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{34} + ( -1 - 6 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{35} + ( 6 - 6 \zeta_{24}^{4} ) q^{36} + ( 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{38} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{39} + ( 6 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{40} + ( 2 - \zeta_{24}^{4} ) q^{41} -6 \zeta_{24}^{6} q^{42} -9 \zeta_{24}^{2} q^{43} + ( -6 + 12 \zeta_{24}^{4} ) q^{44} + ( 3 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{45} + 4 q^{46} + ( -5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{47} + ( 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{48} + \zeta_{24}^{4} q^{49} + ( \zeta_{24} - 7 \zeta_{24}^{7} ) q^{50} + ( 9 - 9 \zeta_{24}^{4} ) q^{51} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{52} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{53} + ( 6 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{54} + ( 6 \zeta_{24} - 3 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{55} + ( 4 + 4 \zeta_{24}^{4} ) q^{56} + ( -5 \zeta_{24}^{2} + 10 \zeta_{24}^{6} ) q^{57} + ( -12 \zeta_{24}^{2} + 12 \zeta_{24}^{6} ) q^{58} + ( -2 + \zeta_{24}^{4} ) q^{59} + ( -2 \zeta_{24} + 8 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{60} + ( -6 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{61} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{62} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{63} -8 q^{64} + ( 1 - 3 \zeta_{24}^{2} + \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{65} + ( -9 \zeta_{24} - 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} ) q^{66} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{67} + ( 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{68} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{69} + ( 8 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{70} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{71} + ( -6 \zeta_{24} + 6 \zeta_{24}^{7} ) q^{72} + 3 \zeta_{24}^{6} q^{73} + ( 6 + 4 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{75} -10 \zeta_{24}^{4} q^{76} + ( -9 \zeta_{24} + 9 \zeta_{24}^{7} ) q^{77} -6 \zeta_{24}^{2} q^{78} + ( -6 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{79} + ( -8 \zeta_{24} - 4 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{80} + ( -9 + 9 \zeta_{24}^{4} ) q^{81} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{82} + ( -6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{84} + ( -3 \zeta_{24} + 12 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{85} + ( 9 \zeta_{24} + 9 \zeta_{24}^{7} ) q^{86} + ( 6 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{87} + ( 12 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{88} + ( 6 - 12 \zeta_{24}^{4} ) q^{89} + ( -9 \zeta_{24}^{2} + 3 \zeta_{24}^{4} + 9 \zeta_{24}^{6} ) q^{90} -6 q^{91} + ( -4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{92} + ( 9 \zeta_{24} - 9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} ) q^{93} + 10 \zeta_{24}^{4} q^{94} + ( -10 \zeta_{24} - 5 \zeta_{24}^{7} ) q^{95} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{96} + 3 \zeta_{24}^{2} q^{97} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{98} + ( 9 - 18 \zeta_{24}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} - 12q^{9} + O(q^{10}) \) \( 8q + 8q^{4} - 12q^{9} - 8q^{10} + 36q^{11} + 24q^{14} - 16q^{16} - 40q^{19} - 16q^{25} - 36q^{30} + 24q^{36} - 8q^{40} + 12q^{41} + 32q^{46} + 4q^{49} + 36q^{51} + 48q^{56} - 12q^{59} - 64q^{64} + 12q^{65} + 36q^{75} - 40q^{76} - 36q^{81} + 12q^{90} - 48q^{91} + 40q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{24}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
0.258819 + 0.965926i
0.965926 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
−1.22474 + 0.707107i −0.866025 1.50000i 1.00000 1.73205i −0.448288 + 2.19067i 2.12132 + 1.22474i −1.22474 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 3.00000i
59.2 −1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 1.73205i 1.67303 1.48356i −2.12132 1.22474i −1.22474 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 + 3.00000i
59.3 1.22474 0.707107i −0.866025 1.50000i 1.00000 1.73205i 0.448288 2.19067i −2.12132 1.22474i 1.22474 + 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 3.00000i
59.4 1.22474 0.707107i 0.866025 + 1.50000i 1.00000 1.73205i −1.67303 + 1.48356i 2.12132 + 1.22474i 1.22474 + 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 + 3.00000i
299.1 −1.22474 0.707107i −0.866025 + 1.50000i 1.00000 + 1.73205i −0.448288 2.19067i 2.12132 1.22474i −1.22474 + 2.12132i 2.82843i −1.50000 2.59808i −1.00000 + 3.00000i
299.2 −1.22474 0.707107i 0.866025 1.50000i 1.00000 + 1.73205i 1.67303 + 1.48356i −2.12132 + 1.22474i −1.22474 + 2.12132i 2.82843i −1.50000 2.59808i −1.00000 3.00000i
299.3 1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 + 1.73205i 0.448288 + 2.19067i −2.12132 + 1.22474i 1.22474 2.12132i 2.82843i −1.50000 2.59808i −1.00000 + 3.00000i
299.4 1.22474 + 0.707107i 0.866025 1.50000i 1.00000 + 1.73205i −1.67303 1.48356i 2.12132 1.22474i 1.22474 2.12132i 2.82843i −1.50000 2.59808i −1.00000 3.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
9.d odd 6 1 inner
40.e odd 2 1 inner
45.h odd 6 1 inner
72.l even 6 1 inner
360.bd even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bd.a 8
5.b even 2 1 inner 360.2.bd.a 8
8.d odd 2 1 inner 360.2.bd.a 8
9.d odd 6 1 inner 360.2.bd.a 8
40.e odd 2 1 inner 360.2.bd.a 8
45.h odd 6 1 inner 360.2.bd.a 8
72.l even 6 1 inner 360.2.bd.a 8
360.bd even 6 1 inner 360.2.bd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bd.a 8 1.a even 1 1 trivial
360.2.bd.a 8 5.b even 2 1 inner
360.2.bd.a 8 8.d odd 2 1 inner
360.2.bd.a 8 9.d odd 6 1 inner
360.2.bd.a 8 40.e odd 2 1 inner
360.2.bd.a 8 45.h odd 6 1 inner
360.2.bd.a 8 72.l even 6 1 inner
360.2.bd.a 8 360.bd even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 6 T_{7}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$5$ \( 625 + 200 T^{2} + 39 T^{4} + 8 T^{6} + T^{8} \)
$7$ \( ( 36 + 6 T^{2} + T^{4} )^{2} \)
$11$ \( ( 27 - 9 T + T^{2} )^{4} \)
$13$ \( ( 36 + 6 T^{2} + T^{4} )^{2} \)
$17$ \( ( -27 + T^{2} )^{4} \)
$19$ \( ( 5 + T )^{8} \)
$23$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$29$ \( ( 5184 + 72 T^{2} + T^{4} )^{2} \)
$31$ \( ( 2916 - 54 T^{2} + T^{4} )^{2} \)
$37$ \( T^{8} \)
$41$ \( ( 3 - 3 T + T^{2} )^{4} \)
$43$ \( ( 6561 - 81 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2500 - 50 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2 + T^{2} )^{4} \)
$59$ \( ( 3 + 3 T + T^{2} )^{4} \)
$61$ \( ( 2916 - 54 T^{2} + T^{4} )^{2} \)
$67$ \( ( 81 - 9 T^{2} + T^{4} )^{2} \)
$71$ \( ( -18 + T^{2} )^{4} \)
$73$ \( ( 9 + T^{2} )^{4} \)
$79$ \( ( 2916 - 54 T^{2} + T^{4} )^{2} \)
$83$ \( T^{8} \)
$89$ \( ( 108 + T^{2} )^{4} \)
$97$ \( ( 81 - 9 T^{2} + T^{4} )^{2} \)
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