Properties

Label 360.2.w.a
Level $360$
Weight $2$
Character orbit 360.w
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.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( 1 - \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( 1 - \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 1 - 3 \zeta_{8}^{2} ) q^{10} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} + ( -2 - 2 \zeta_{8}^{2} ) q^{13} + 2 \zeta_{8} q^{14} + 4 q^{16} -4 \zeta_{8} q^{17} -6 \zeta_{8}^{2} q^{19} + ( 4 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{20} -6 \zeta_{8}^{2} q^{22} + 2 \zeta_{8} q^{23} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} -4 \zeta_{8}^{3} q^{26} + ( -2 + 2 \zeta_{8}^{2} ) q^{28} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} -6 \zeta_{8}^{2} q^{31} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + ( 4 - 4 \zeta_{8}^{2} ) q^{34} + ( -\zeta_{8} + 3 \zeta_{8}^{3} ) q^{35} + ( -8 + 8 \zeta_{8}^{2} ) q^{37} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{38} + ( -2 + 6 \zeta_{8}^{2} ) q^{40} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{41} + ( -2 + 2 \zeta_{8}^{2} ) q^{43} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{44} + ( -2 + 2 \zeta_{8}^{2} ) q^{46} + 2 \zeta_{8}^{3} q^{47} + 5 \zeta_{8}^{2} q^{49} + ( \zeta_{8} + 7 \zeta_{8}^{3} ) q^{50} + ( 4 + 4 \zeta_{8}^{2} ) q^{52} + 2 \zeta_{8} q^{53} + ( 9 + 3 \zeta_{8}^{2} ) q^{55} -4 \zeta_{8} q^{56} + 6 \zeta_{8}^{2} q^{58} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{59} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{62} -8 q^{64} + ( 6 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{65} + ( -8 - 8 \zeta_{8}^{2} ) q^{67} + 8 \zeta_{8} q^{68} + ( -2 - 4 \zeta_{8}^{2} ) q^{70} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{71} + ( -5 + 5 \zeta_{8}^{2} ) q^{73} -16 \zeta_{8} q^{74} + 12 \zeta_{8}^{2} q^{76} + 6 \zeta_{8}^{3} q^{77} + 14 q^{79} + ( -8 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{80} -12 \zeta_{8}^{2} q^{82} + 2 \zeta_{8}^{3} q^{83} + ( 4 + 8 \zeta_{8}^{2} ) q^{85} -4 \zeta_{8} q^{86} + 12 \zeta_{8}^{2} q^{88} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{89} -4 q^{91} -4 \zeta_{8} q^{92} + ( -2 - 2 \zeta_{8}^{2} ) q^{94} + ( 6 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{95} + ( 7 + 7 \zeta_{8}^{2} ) q^{97} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + 4q^{7} + O(q^{10}) \) \( 4q - 8q^{4} + 4q^{7} + 4q^{10} - 8q^{13} + 16q^{16} + 16q^{25} - 8q^{28} + 16q^{34} - 32q^{37} - 8q^{40} - 8q^{43} - 8q^{46} + 16q^{52} + 36q^{55} - 32q^{64} - 32q^{67} - 8q^{70} - 20q^{73} + 56q^{79} + 16q^{85} - 16q^{91} - 8q^{94} + 28q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(\zeta_{8}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.41421i 0 −2.00000 −2.12132 + 0.707107i 0 1.00000 + 1.00000i 2.82843i 0 1.00000 + 3.00000i
163.2 1.41421i 0 −2.00000 2.12132 0.707107i 0 1.00000 + 1.00000i 2.82843i 0 1.00000 + 3.00000i
307.1 1.41421i 0 −2.00000 2.12132 + 0.707107i 0 1.00000 1.00000i 2.82843i 0 1.00000 3.00000i
307.2 1.41421i 0 −2.00000 −2.12132 0.707107i 0 1.00000 1.00000i 2.82843i 0 1.00000 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.w.a 4
3.b odd 2 1 inner 360.2.w.a 4
4.b odd 2 1 1440.2.bi.a 4
5.c odd 4 1 360.2.w.b yes 4
8.b even 2 1 1440.2.bi.b 4
8.d odd 2 1 360.2.w.b yes 4
12.b even 2 1 1440.2.bi.a 4
15.e even 4 1 360.2.w.b yes 4
20.e even 4 1 1440.2.bi.b 4
24.f even 2 1 360.2.w.b yes 4
24.h odd 2 1 1440.2.bi.b 4
40.i odd 4 1 1440.2.bi.a 4
40.k even 4 1 inner 360.2.w.a 4
60.l odd 4 1 1440.2.bi.b 4
120.q odd 4 1 inner 360.2.w.a 4
120.w even 4 1 1440.2.bi.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.w.a 4 1.a even 1 1 trivial
360.2.w.a 4 3.b odd 2 1 inner
360.2.w.a 4 40.k even 4 1 inner
360.2.w.a 4 120.q odd 4 1 inner
360.2.w.b yes 4 5.c odd 4 1
360.2.w.b yes 4 8.d odd 2 1
360.2.w.b yes 4 15.e even 4 1
360.2.w.b yes 4 24.f even 2 1
1440.2.bi.a 4 4.b odd 2 1
1440.2.bi.a 4 12.b even 2 1
1440.2.bi.a 4 40.i odd 4 1
1440.2.bi.a 4 120.w even 4 1
1440.2.bi.b 4 8.b even 2 1
1440.2.bi.b 4 20.e even 4 1
1440.2.bi.b 4 24.h odd 2 1
1440.2.bi.b 4 60.l odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 25 - 8 T^{2} + T^{4} \)
$7$ \( ( 2 - 2 T + T^{2} )^{2} \)
$11$ \( ( -18 + T^{2} )^{2} \)
$13$ \( ( 8 + 4 T + T^{2} )^{2} \)
$17$ \( 256 + T^{4} \)
$19$ \( ( 36 + T^{2} )^{2} \)
$23$ \( 16 + T^{4} \)
$29$ \( ( -18 + T^{2} )^{2} \)
$31$ \( ( 36 + T^{2} )^{2} \)
$37$ \( ( 128 + 16 T + T^{2} )^{2} \)
$41$ \( ( -72 + T^{2} )^{2} \)
$43$ \( ( 8 + 4 T + T^{2} )^{2} \)
$47$ \( 16 + T^{4} \)
$53$ \( 16 + T^{4} \)
$59$ \( ( 98 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 128 + 16 T + T^{2} )^{2} \)
$71$ \( ( 200 + T^{2} )^{2} \)
$73$ \( ( 50 + 10 T + T^{2} )^{2} \)
$79$ \( ( -14 + T )^{4} \)
$83$ \( 16 + T^{4} \)
$89$ \( ( 8 + T^{2} )^{2} \)
$97$ \( ( 98 - 14 T + T^{2} )^{2} \)
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