Properties

Label 360.2.bf.a
Level $360$
Weight $2$
Character orbit 360.bf
Analytic conductor $2.875$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + \zeta_{12} q^{5} + ( 1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + 4 \zeta_{12}^{2} q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + \zeta_{12} q^{5} + ( 1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + 4 \zeta_{12}^{2} q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -1 - \zeta_{12}^{3} ) q^{10} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{12} -6 \zeta_{12} q^{13} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{14} + ( 1 - 2 \zeta_{12}^{2} ) q^{15} + 4 \zeta_{12}^{2} q^{16} -3 q^{17} + ( 3 - 3 \zeta_{12}^{3} ) q^{18} + 7 \zeta_{12}^{3} q^{19} + 2 \zeta_{12}^{2} q^{20} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{21} + ( 5 + 5 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{22} + ( -2 + 2 \zeta_{12}^{2} ) q^{23} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{24} + \zeta_{12}^{2} q^{25} + ( 6 + 6 \zeta_{12}^{3} ) q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + 8 \zeta_{12}^{3} q^{28} + ( -2 + \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{30} + ( -4 + 4 \zeta_{12}^{2} ) q^{31} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32} + ( 5 + 5 \zeta_{12}^{2} ) q^{33} + ( 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{34} + 4 \zeta_{12}^{3} q^{35} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{36} -2 \zeta_{12}^{3} q^{37} + ( 7 \zeta_{12} - 7 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{38} + ( -6 + 12 \zeta_{12}^{2} ) q^{39} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{40} + ( 5 - 5 \zeta_{12}^{2} ) q^{41} + ( -4 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{42} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{43} -10 q^{44} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{45} + ( 2 - 2 \zeta_{12}^{3} ) q^{46} -2 \zeta_{12}^{2} q^{47} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{48} + ( -9 + 9 \zeta_{12}^{2} ) q^{49} + ( 1 - \zeta_{12} - \zeta_{12}^{2} ) q^{50} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{51} -12 \zeta_{12}^{2} q^{52} + 8 \zeta_{12}^{3} q^{53} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{54} -5 q^{55} + ( 8 \zeta_{12} - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{56} + ( 14 - 7 \zeta_{12}^{2} ) q^{57} + 7 \zeta_{12} q^{59} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{60} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{61} + ( 4 - 4 \zeta_{12}^{3} ) q^{62} -12 q^{63} + 8 \zeta_{12}^{3} q^{64} -6 \zeta_{12}^{2} q^{65} + ( 5 - 10 \zeta_{12} - 10 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{66} + 3 \zeta_{12} q^{67} -6 \zeta_{12} q^{68} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{69} + ( 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{70} + 2 q^{71} + ( 6 + 6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{72} + q^{73} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{74} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{75} + ( -14 + 14 \zeta_{12}^{2} ) q^{76} -20 \zeta_{12} q^{77} + ( 12 - 6 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{78} -10 \zeta_{12}^{2} q^{79} + 4 \zeta_{12}^{3} q^{80} -9 \zeta_{12}^{2} q^{81} + ( -5 + 5 \zeta_{12}^{3} ) q^{82} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{83} + ( 16 - 8 \zeta_{12}^{2} ) q^{84} -3 \zeta_{12} q^{85} + ( -5 - 5 \zeta_{12} + 5 \zeta_{12}^{2} ) q^{86} + ( 10 \zeta_{12} + 10 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{88} + 2 q^{89} + ( 3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{90} -24 \zeta_{12}^{3} q^{91} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{92} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{94} + ( -7 + 7 \zeta_{12}^{2} ) q^{95} + ( -4 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{96} + 7 \zeta_{12}^{2} q^{97} + ( 9 - 9 \zeta_{12}^{3} ) q^{98} -15 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 6q^{6} + 8q^{7} - 8q^{8} - 6q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 6q^{6} + 8q^{7} - 8q^{8} - 6q^{9} - 4q^{10} + 8q^{14} + 8q^{16} - 12q^{17} + 12q^{18} + 4q^{20} + 10q^{22} - 4q^{23} - 12q^{24} + 2q^{25} + 24q^{26} - 6q^{30} - 8q^{31} + 8q^{32} + 30q^{33} + 6q^{34} - 14q^{38} + 4q^{40} + 10q^{41} - 40q^{44} + 8q^{46} - 4q^{47} - 18q^{49} + 2q^{50} - 24q^{52} - 18q^{54} - 20q^{55} - 16q^{56} + 42q^{57} + 16q^{62} - 48q^{63} - 12q^{65} - 8q^{70} + 8q^{71} + 12q^{72} + 4q^{73} + 4q^{74} - 28q^{76} + 36q^{78} - 20q^{79} - 18q^{81} - 20q^{82} + 48q^{84} - 10q^{86} + 20q^{88} + 8q^{89} + 6q^{90} - 4q^{94} - 14q^{95} + 14q^{97} + 36q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−1.36603 + 0.366025i −0.866025 + 1.50000i 1.73205 1.00000i 0.866025 0.500000i 0.633975 2.36603i 2.00000 3.46410i −2.00000 + 2.00000i −1.50000 2.59808i −1.00000 + 1.00000i
61.2 0.366025 + 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i −0.866025 + 0.500000i 2.36603 + 0.633975i 2.00000 3.46410i −2.00000 2.00000i −1.50000 2.59808i −1.00000 1.00000i
301.1 −1.36603 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i 0.866025 + 0.500000i 0.633975 + 2.36603i 2.00000 + 3.46410i −2.00000 2.00000i −1.50000 + 2.59808i −1.00000 1.00000i
301.2 0.366025 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i −0.866025 0.500000i 2.36603 0.633975i 2.00000 + 3.46410i −2.00000 + 2.00000i −1.50000 + 2.59808i −1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n 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.bf.a 4
3.b odd 2 1 1080.2.bf.a 4
4.b odd 2 1 1440.2.bv.a 4
8.b even 2 1 inner 360.2.bf.a 4
8.d odd 2 1 1440.2.bv.a 4
9.c even 3 1 inner 360.2.bf.a 4
9.d odd 6 1 1080.2.bf.a 4
12.b even 2 1 4320.2.bv.a 4
24.f even 2 1 4320.2.bv.a 4
24.h odd 2 1 1080.2.bf.a 4
36.f odd 6 1 1440.2.bv.a 4
36.h even 6 1 4320.2.bv.a 4
72.j odd 6 1 1080.2.bf.a 4
72.l even 6 1 4320.2.bv.a 4
72.n even 6 1 inner 360.2.bf.a 4
72.p odd 6 1 1440.2.bv.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bf.a 4 1.a even 1 1 trivial
360.2.bf.a 4 8.b even 2 1 inner
360.2.bf.a 4 9.c even 3 1 inner
360.2.bf.a 4 72.n even 6 1 inner
1080.2.bf.a 4 3.b odd 2 1
1080.2.bf.a 4 9.d odd 6 1
1080.2.bf.a 4 24.h odd 2 1
1080.2.bf.a 4 72.j odd 6 1
1440.2.bv.a 4 4.b odd 2 1
1440.2.bv.a 4 8.d odd 2 1
1440.2.bv.a 4 36.f odd 6 1
1440.2.bv.a 4 72.p odd 6 1
4320.2.bv.a 4 12.b even 2 1
4320.2.bv.a 4 24.f even 2 1
4320.2.bv.a 4 36.h even 6 1
4320.2.bv.a 4 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 4 T_{7} + 16 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( ( 16 - 4 T + T^{2} )^{2} \)
$11$ \( 625 - 25 T^{2} + T^{4} \)
$13$ \( 1296 - 36 T^{2} + T^{4} \)
$17$ \( ( 3 + T )^{4} \)
$19$ \( ( 49 + T^{2} )^{2} \)
$23$ \( ( 4 + 2 T + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 16 + 4 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 25 - 5 T + T^{2} )^{2} \)
$43$ \( 625 - 25 T^{2} + T^{4} \)
$47$ \( ( 4 + 2 T + T^{2} )^{2} \)
$53$ \( ( 64 + T^{2} )^{2} \)
$59$ \( 2401 - 49 T^{2} + T^{4} \)
$61$ \( 256 - 16 T^{2} + T^{4} \)
$67$ \( 81 - 9 T^{2} + T^{4} \)
$71$ \( ( -2 + T )^{4} \)
$73$ \( ( -1 + T )^{4} \)
$79$ \( ( 100 + 10 T + T^{2} )^{2} \)
$83$ \( 20736 - 144 T^{2} + T^{4} \)
$89$ \( ( -2 + T )^{4} \)
$97$ \( ( 49 - 7 T + T^{2} )^{2} \)
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