Properties

Label 360.2.w.c
Level $360$
Weight $2$
Character orbit 360.w
Analytic conductor $2.875$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 40)
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 \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{2} + 2 \zeta_{20} q^{4} + ( -\zeta_{20}^{3} + \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{5} + ( -2 \zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{7} + ( 2 \zeta_{20} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{2} + 2 \zeta_{20} q^{4} + ( -\zeta_{20}^{3} + \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{5} + ( -2 \zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{7} + ( 2 \zeta_{20} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{8} + ( -1 - 2 \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{10} + ( 2 + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{11} + ( 1 + 2 \zeta_{20} - 2 \zeta_{20}^{2} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} + \zeta_{20}^{5} ) q^{13} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{14} + 4 \zeta_{20}^{2} q^{16} + ( 1 + \zeta_{20}^{5} ) q^{17} -2 \zeta_{20}^{5} q^{19} + ( -2 + 2 \zeta_{20}^{2} - 4 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{20} + ( 2 + 2 \zeta_{20} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} ) q^{22} + ( 2 - 2 \zeta_{20} - 4 \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} - 3 \zeta_{20}^{6} + 3 \zeta_{20}^{7} ) q^{23} + ( -1 - 4 \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{25} + ( -2 + 2 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{26} + ( -2 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{28} + ( 4 \zeta_{20} + 2 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{29} + ( 4 - 8 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{31} + ( 4 + 4 \zeta_{20}^{5} ) q^{32} + 2 \zeta_{20}^{3} q^{34} + ( -2 - 3 \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{6} - 3 \zeta_{20}^{7} ) q^{35} + ( 1 - 2 \zeta_{20} - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} - \zeta_{20}^{5} ) q^{37} + ( 2 - 2 \zeta_{20}^{2} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{38} + ( -4 - 2 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 4 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{40} + ( -2 - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{6} ) q^{41} + ( 4 - \zeta_{20}^{3} + \zeta_{20}^{4} - 3 \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{43} + ( 4 \zeta_{20} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{44} + ( -4 - 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{46} + ( 2 - 6 \zeta_{20} - 4 \zeta_{20}^{2} + 5 \zeta_{20}^{3} + 5 \zeta_{20}^{4} - 3 \zeta_{20}^{5} + \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{47} + ( -2 \zeta_{20}^{3} + 3 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{49} + ( 1 - 6 \zeta_{20} + \zeta_{20}^{2} + 3 \zeta_{20}^{3} - \zeta_{20}^{4} - 6 \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{50} + ( 2 \zeta_{20} + 4 \zeta_{20}^{2} - 4 \zeta_{20}^{3} - 4 \zeta_{20}^{4} + 4 \zeta_{20}^{5} + 2 \zeta_{20}^{6} ) q^{52} + ( -1 + 2 \zeta_{20} + 2 \zeta_{20}^{2} + \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{53} + ( -2 - 4 \zeta_{20} + 4 \zeta_{20}^{2} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 4 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{55} + ( -8 + 4 \zeta_{20}^{2} - 4 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{56} + ( 2 + 4 \zeta_{20} + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} + 2 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{58} + ( 4 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{59} + ( -4 + 8 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 2 \zeta_{20}^{6} ) q^{61} + ( -4 + 6 \zeta_{20} - 2 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{5} - 4 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{62} + 8 \zeta_{20}^{3} q^{64} + ( -1 - 4 \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 7 \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{65} + ( -4 - \zeta_{20}^{3} - \zeta_{20}^{4} - 3 \zeta_{20}^{5} + \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{67} + ( 2 \zeta_{20} + 2 \zeta_{20}^{6} ) q^{68} + ( 2 - 2 \zeta_{20} + 2 \zeta_{20}^{2} - 4 \zeta_{20}^{3} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} ) q^{70} + ( -6 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{71} + ( -1 + 6 \zeta_{20}^{3} - 6 \zeta_{20}^{4} - 5 \zeta_{20}^{5} + 6 \zeta_{20}^{6} + 6 \zeta_{20}^{7} ) q^{73} + ( 2 \zeta_{20}^{3} - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{74} -4 \zeta_{20}^{6} q^{76} + ( -2 - 4 \zeta_{20} + 4 \zeta_{20}^{2} - 2 \zeta_{20}^{5} + 4 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{77} + ( 8 \zeta_{20} - 8 \zeta_{20}^{3} + 4 \zeta_{20}^{5} ) q^{79} + ( -4 - 4 \zeta_{20} + 4 \zeta_{20}^{2} + 4 \zeta_{20}^{3} - 4 \zeta_{20}^{4} - 8 \zeta_{20}^{5} + 8 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{80} + ( -2 - 6 \zeta_{20} - 4 \zeta_{20}^{2} + 4 \zeta_{20}^{3} + 4 \zeta_{20}^{4} - 6 \zeta_{20}^{5} + 2 \zeta_{20}^{6} ) q^{82} + ( 6 - \zeta_{20}^{3} + \zeta_{20}^{4} - 5 \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{83} + ( 1 - 2 \zeta_{20} - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{85} + ( 8 - 6 \zeta_{20}^{2} + 6 \zeta_{20}^{4} - 8 \zeta_{20}^{6} ) q^{86} + ( 4 \zeta_{20} + 4 \zeta_{20}^{2} + 4 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{88} + ( -4 \zeta_{20}^{3} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{89} + ( -4 + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{91} + ( -6 + 4 \zeta_{20} + 2 \zeta_{20}^{2} - 8 \zeta_{20}^{3} - 8 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 4 \zeta_{20}^{6} - 6 \zeta_{20}^{7} ) q^{92} + ( -2 \zeta_{20} + 6 \zeta_{20}^{3} - 10 \zeta_{20}^{5} + 12 \zeta_{20}^{7} ) q^{94} + ( -2 + 4 \zeta_{20} + 4 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{95} + ( 3 + 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} + \zeta_{20}^{5} - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{97} + ( -1 - 2 \zeta_{20} + 3 \zeta_{20}^{2} + 3 \zeta_{20}^{3} - 3 \zeta_{20}^{4} - 2 \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} - 4q^{8} + O(q^{10}) \) \( 8q + 2q^{2} - 4q^{8} - 10q^{10} + 8q^{11} + 8q^{16} + 8q^{17} + 12q^{22} - 20q^{26} - 20q^{28} + 32q^{32} - 20q^{35} + 4q^{38} - 20q^{40} + 8q^{41} + 28q^{43} - 40q^{46} + 10q^{50} + 20q^{52} - 40q^{56} + 20q^{58} - 40q^{62} - 28q^{67} + 4q^{68} + 20q^{70} + 16q^{73} - 8q^{76} - 28q^{82} + 44q^{83} + 24q^{86} + 16q^{88} - 40q^{91} - 20q^{92} + 16q^{97} + 6q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(\zeta_{20}^{3}\) \(-1\) \(1\)

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
0.587785 + 0.809017i
−0.951057 0.309017i
−0.587785 + 0.809017i
0.951057 0.309017i
0.587785 0.809017i
−0.951057 + 0.309017i
−0.587785 0.809017i
0.951057 + 0.309017i
−1.26007 0.642040i 0 1.17557 + 1.61803i 1.90211 1.17557i 0 −1.17557 1.17557i −0.442463 2.79360i 0 −3.15156 + 0.260074i
163.2 0.221232 1.39680i 0 −1.90211 0.618034i 1.17557 + 1.90211i 0 1.90211 + 1.90211i −1.28408 + 2.52015i 0 2.91695 1.22123i
163.3 0.642040 + 1.26007i 0 −1.17557 + 1.61803i −1.90211 + 1.17557i 0 1.17557 + 1.17557i −2.79360 0.442463i 0 −2.70254 1.64204i
163.4 1.39680 0.221232i 0 1.90211 0.618034i −1.17557 1.90211i 0 −1.90211 1.90211i 2.52015 1.28408i 0 −2.06285 2.39680i
307.1 −1.26007 + 0.642040i 0 1.17557 1.61803i 1.90211 + 1.17557i 0 −1.17557 + 1.17557i −0.442463 + 2.79360i 0 −3.15156 0.260074i
307.2 0.221232 + 1.39680i 0 −1.90211 + 0.618034i 1.17557 1.90211i 0 1.90211 1.90211i −1.28408 2.52015i 0 2.91695 + 1.22123i
307.3 0.642040 1.26007i 0 −1.17557 1.61803i −1.90211 1.17557i 0 1.17557 1.17557i −2.79360 + 0.442463i 0 −2.70254 + 1.64204i
307.4 1.39680 + 0.221232i 0 1.90211 + 0.618034i −1.17557 + 1.90211i 0 −1.90211 + 1.90211i 2.52015 + 1.28408i 0 −2.06285 + 2.39680i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 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.c 8
3.b odd 2 1 40.2.k.a 8
4.b odd 2 1 1440.2.bi.c 8
5.c odd 4 1 inner 360.2.w.c 8
8.b even 2 1 1440.2.bi.c 8
8.d odd 2 1 inner 360.2.w.c 8
12.b even 2 1 160.2.o.a 8
15.d odd 2 1 200.2.k.h 8
15.e even 4 1 40.2.k.a 8
15.e even 4 1 200.2.k.h 8
20.e even 4 1 1440.2.bi.c 8
24.f even 2 1 40.2.k.a 8
24.h odd 2 1 160.2.o.a 8
40.i odd 4 1 1440.2.bi.c 8
40.k even 4 1 inner 360.2.w.c 8
48.i odd 4 1 1280.2.n.m 8
48.i odd 4 1 1280.2.n.q 8
48.k even 4 1 1280.2.n.m 8
48.k even 4 1 1280.2.n.q 8
60.h even 2 1 800.2.o.g 8
60.l odd 4 1 160.2.o.a 8
60.l odd 4 1 800.2.o.g 8
120.i odd 2 1 800.2.o.g 8
120.m even 2 1 200.2.k.h 8
120.q odd 4 1 40.2.k.a 8
120.q odd 4 1 200.2.k.h 8
120.w even 4 1 160.2.o.a 8
120.w even 4 1 800.2.o.g 8
240.z odd 4 1 1280.2.n.m 8
240.bb even 4 1 1280.2.n.q 8
240.bd odd 4 1 1280.2.n.q 8
240.bf even 4 1 1280.2.n.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.k.a 8 3.b odd 2 1
40.2.k.a 8 15.e even 4 1
40.2.k.a 8 24.f even 2 1
40.2.k.a 8 120.q odd 4 1
160.2.o.a 8 12.b even 2 1
160.2.o.a 8 24.h odd 2 1
160.2.o.a 8 60.l odd 4 1
160.2.o.a 8 120.w even 4 1
200.2.k.h 8 15.d odd 2 1
200.2.k.h 8 15.e even 4 1
200.2.k.h 8 120.m even 2 1
200.2.k.h 8 120.q odd 4 1
360.2.w.c 8 1.a even 1 1 trivial
360.2.w.c 8 5.c odd 4 1 inner
360.2.w.c 8 8.d odd 2 1 inner
360.2.w.c 8 40.k even 4 1 inner
800.2.o.g 8 60.h even 2 1
800.2.o.g 8 60.l odd 4 1
800.2.o.g 8 120.i odd 2 1
800.2.o.g 8 120.w even 4 1
1280.2.n.m 8 48.i odd 4 1
1280.2.n.m 8 48.k even 4 1
1280.2.n.m 8 240.z odd 4 1
1280.2.n.m 8 240.bf even 4 1
1280.2.n.q 8 48.i odd 4 1
1280.2.n.q 8 48.k even 4 1
1280.2.n.q 8 240.bb even 4 1
1280.2.n.q 8 240.bd odd 4 1
1440.2.bi.c 8 4.b odd 2 1
1440.2.bi.c 8 8.b even 2 1
1440.2.bi.c 8 20.e even 4 1
1440.2.bi.c 8 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 60 T_{7}^{4} + 400 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 625 + 30 T^{4} + T^{8} \)
$7$ \( 400 + 60 T^{4} + T^{8} \)
$11$ \( ( -4 - 2 T + T^{2} )^{4} \)
$13$ \( 400 + 360 T^{4} + T^{8} \)
$17$ \( ( 2 - 2 T + T^{2} )^{4} \)
$19$ \( ( 4 + T^{2} )^{4} \)
$23$ \( 250000 + 1500 T^{4} + T^{8} \)
$29$ \( ( 80 - 40 T^{2} + T^{4} )^{2} \)
$31$ \( ( 2000 + 100 T^{2} + T^{4} )^{2} \)
$37$ \( 400 + 360 T^{4} + T^{8} \)
$41$ \( ( -44 - 2 T + T^{2} )^{4} \)
$43$ \( ( 484 - 308 T + 98 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$47$ \( 5856400 + 12060 T^{4} + T^{8} \)
$53$ \( 400 + 360 T^{4} + T^{8} \)
$59$ \( ( 16 + 72 T^{2} + T^{4} )^{2} \)
$61$ \( ( 80 + 100 T^{2} + T^{4} )^{2} \)
$67$ \( ( 484 + 308 T + 98 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$71$ \( ( 6480 + 180 T^{2} + T^{4} )^{2} \)
$73$ \( ( 6724 + 656 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$79$ \( ( 1280 - 160 T^{2} + T^{4} )^{2} \)
$83$ \( ( 3364 - 1276 T + 242 T^{2} - 22 T^{3} + T^{4} )^{2} \)
$89$ \( ( 256 + 48 T^{2} + T^{4} )^{2} \)
$97$ \( ( 4 + 16 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
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