Properties

Label 360.2.bi.a
Level $360$
Weight $2$
Character orbit 360.bi
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.bi (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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} -\zeta_{12} q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} -\zeta_{12} q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -2 + 2 \zeta_{12}^{2} ) q^{11} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( 2 + \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{15} + 6 \zeta_{12}^{3} q^{17} -2 q^{19} + ( -1 + 2 \zeta_{12}^{2} ) q^{21} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{23} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -7 + 7 \zeta_{12}^{2} ) q^{29} + 6 \zeta_{12}^{2} q^{31} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{33} + ( 2 - \zeta_{12}^{3} ) q^{35} + 2 \zeta_{12}^{3} q^{37} + ( 2 + 2 \zeta_{12}^{2} ) q^{39} -5 \zeta_{12}^{2} q^{41} -12 \zeta_{12} q^{43} + ( -3 - 6 \zeta_{12}^{3} ) q^{45} + 9 \zeta_{12} q^{47} -6 \zeta_{12}^{2} q^{49} + ( 12 - 6 \zeta_{12}^{2} ) q^{51} + 8 \zeta_{12}^{3} q^{53} + ( -2 - 4 \zeta_{12}^{3} ) q^{55} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} -12 \zeta_{12}^{2} q^{59} + ( 7 - 7 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{65} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{67} + ( -1 - \zeta_{12}^{2} ) q^{69} -10 q^{71} -4 \zeta_{12}^{3} q^{73} + ( -4 - 6 \zeta_{12} + 8 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{75} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} + ( 4 - 4 \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} -5 \zeta_{12} q^{83} + ( -6 \zeta_{12} - 12 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{85} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{87} + 15 q^{89} + 2 q^{91} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{93} + ( 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{95} + 16 \zeta_{12} q^{97} -6 \zeta_{12}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{5} - 6q^{9} - 4q^{11} + 12q^{15} - 8q^{19} + 6q^{25} - 14q^{29} + 12q^{31} + 8q^{35} + 12q^{39} - 10q^{41} - 12q^{45} - 12q^{49} + 36q^{51} - 8q^{55} - 24q^{59} + 14q^{61} + 8q^{65} - 6q^{69} - 40q^{71} + 8q^{79} - 18q^{81} - 24q^{85} + 60q^{89} + 8q^{91} - 4q^{95} - 12q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 1.50000i 0 −1.23205 1.86603i 0 −0.866025 + 0.500000i 0 −1.50000 2.59808i 0
49.2 0 0.866025 1.50000i 0 2.23205 + 0.133975i 0 0.866025 0.500000i 0 −1.50000 2.59808i 0
169.1 0 −0.866025 1.50000i 0 −1.23205 + 1.86603i 0 −0.866025 0.500000i 0 −1.50000 + 2.59808i 0
169.2 0 0.866025 + 1.50000i 0 2.23205 0.133975i 0 0.866025 + 0.500000i 0 −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j 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.bi.a 4
3.b odd 2 1 1080.2.bi.a 4
4.b odd 2 1 720.2.by.b 4
5.b even 2 1 inner 360.2.bi.a 4
9.c even 3 1 inner 360.2.bi.a 4
9.c even 3 1 3240.2.f.b 2
9.d odd 6 1 1080.2.bi.a 4
9.d odd 6 1 3240.2.f.e 2
12.b even 2 1 2160.2.by.a 4
15.d odd 2 1 1080.2.bi.a 4
20.d odd 2 1 720.2.by.b 4
36.f odd 6 1 720.2.by.b 4
36.h even 6 1 2160.2.by.a 4
45.h odd 6 1 1080.2.bi.a 4
45.h odd 6 1 3240.2.f.e 2
45.j even 6 1 inner 360.2.bi.a 4
45.j even 6 1 3240.2.f.b 2
60.h even 2 1 2160.2.by.a 4
180.n even 6 1 2160.2.by.a 4
180.p odd 6 1 720.2.by.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bi.a 4 1.a even 1 1 trivial
360.2.bi.a 4 5.b even 2 1 inner
360.2.bi.a 4 9.c even 3 1 inner
360.2.bi.a 4 45.j even 6 1 inner
720.2.by.b 4 4.b odd 2 1
720.2.by.b 4 20.d odd 2 1
720.2.by.b 4 36.f odd 6 1
720.2.by.b 4 180.p odd 6 1
1080.2.bi.a 4 3.b odd 2 1
1080.2.bi.a 4 9.d odd 6 1
1080.2.bi.a 4 15.d odd 2 1
1080.2.bi.a 4 45.h odd 6 1
2160.2.by.a 4 12.b even 2 1
2160.2.by.a 4 36.h even 6 1
2160.2.by.a 4 60.h even 2 1
2160.2.by.a 4 180.n even 6 1
3240.2.f.b 2 9.c even 3 1
3240.2.f.b 2 45.j even 6 1
3240.2.f.e 2 9.d odd 6 1
3240.2.f.e 2 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( 25 - 10 T - T^{2} - 2 T^{3} + T^{4} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( ( 4 + 2 T + T^{2} )^{2} \)
$13$ \( 16 - 4 T^{2} + T^{4} \)
$17$ \( ( 36 + T^{2} )^{2} \)
$19$ \( ( 2 + T )^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 49 + 7 T + T^{2} )^{2} \)
$31$ \( ( 36 - 6 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 25 + 5 T + T^{2} )^{2} \)
$43$ \( 20736 - 144 T^{2} + T^{4} \)
$47$ \( 6561 - 81 T^{2} + T^{4} \)
$53$ \( ( 64 + T^{2} )^{2} \)
$59$ \( ( 144 + 12 T + T^{2} )^{2} \)
$61$ \( ( 49 - 7 T + T^{2} )^{2} \)
$67$ \( 625 - 25 T^{2} + T^{4} \)
$71$ \( ( 10 + T )^{4} \)
$73$ \( ( 16 + T^{2} )^{2} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( 625 - 25 T^{2} + T^{4} \)
$89$ \( ( -15 + T )^{4} \)
$97$ \( 65536 - 256 T^{2} + T^{4} \)
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