Properties

Label 360.1.p.a
Level 360
Weight 1
Character orbit 360.p
Self dual yes
Analytic conductor 0.180
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -15, -40, 24
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 360.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.179663404548\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{6}, \sqrt{-10})\)
Artin image $D_4$
Artin field Galois closure of 4.0.5400.2

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} + q^{16} - 2q^{19} + q^{20} + 2q^{23} + q^{25} - q^{32} + 2q^{38} - q^{40} - 2q^{46} - 2q^{47} - q^{49} - q^{50} - 2q^{53} + q^{64} - 2q^{76} + q^{80} + 2q^{92} + 2q^{94} - 2q^{95} + q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−1.00000 0 1.00000 1.00000 0 0 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.1.p.a 1
3.b odd 2 1 360.1.p.b yes 1
4.b odd 2 1 1440.1.p.b 1
5.b even 2 1 360.1.p.b yes 1
5.c odd 4 2 1800.1.g.c 2
8.b even 2 1 1440.1.p.a 1
8.d odd 2 1 360.1.p.b yes 1
9.c even 3 2 3240.1.z.f 2
9.d odd 6 2 3240.1.z.d 2
12.b even 2 1 1440.1.p.a 1
15.d odd 2 1 CM 360.1.p.a 1
15.e even 4 2 1800.1.g.c 2
20.d odd 2 1 1440.1.p.a 1
24.f even 2 1 RM 360.1.p.a 1
24.h odd 2 1 1440.1.p.b 1
40.e odd 2 1 CM 360.1.p.a 1
40.f even 2 1 1440.1.p.b 1
40.k even 4 2 1800.1.g.c 2
45.h odd 6 2 3240.1.z.f 2
45.j even 6 2 3240.1.z.d 2
60.h even 2 1 1440.1.p.b 1
72.l even 6 2 3240.1.z.f 2
72.p odd 6 2 3240.1.z.d 2
120.i odd 2 1 1440.1.p.a 1
120.m even 2 1 360.1.p.b yes 1
120.q odd 4 2 1800.1.g.c 2
360.z odd 6 2 3240.1.z.f 2
360.bd even 6 2 3240.1.z.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.1.p.a 1 1.a even 1 1 trivial
360.1.p.a 1 15.d odd 2 1 CM
360.1.p.a 1 24.f even 2 1 RM
360.1.p.a 1 40.e odd 2 1 CM
360.1.p.b yes 1 3.b odd 2 1
360.1.p.b yes 1 5.b even 2 1
360.1.p.b yes 1 8.d odd 2 1
360.1.p.b yes 1 120.m even 2 1
1440.1.p.a 1 8.b even 2 1
1440.1.p.a 1 12.b even 2 1
1440.1.p.a 1 20.d odd 2 1
1440.1.p.a 1 120.i odd 2 1
1440.1.p.b 1 4.b odd 2 1
1440.1.p.b 1 24.h odd 2 1
1440.1.p.b 1 40.f even 2 1
1440.1.p.b 1 60.h even 2 1
1800.1.g.c 2 5.c odd 4 2
1800.1.g.c 2 15.e even 4 2
1800.1.g.c 2 40.k even 4 2
1800.1.g.c 2 120.q odd 4 2
3240.1.z.d 2 9.d odd 6 2
3240.1.z.d 2 45.j even 6 2
3240.1.z.d 2 72.p odd 6 2
3240.1.z.d 2 360.bd even 6 2
3240.1.z.f 2 9.c even 3 2
3240.1.z.f 2 45.h odd 6 2
3240.1.z.f 2 72.l even 6 2
3240.1.z.f 2 360.z odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23} - 2 \) acting on \(S_{1}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ 1
$5$ \( 1 - T \)
$7$ \( 1 + T^{2} \)
$11$ \( 1 + T^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( ( 1 - T )^{2} \)
$29$ \( ( 1 - T )( 1 + T ) \)
$31$ \( ( 1 - T )( 1 + T ) \)
$37$ \( 1 + T^{2} \)
$41$ \( 1 + T^{2} \)
$43$ \( ( 1 - T )( 1 + T ) \)
$47$ \( ( 1 + T )^{2} \)
$53$ \( ( 1 + T )^{2} \)
$59$ \( 1 + T^{2} \)
$61$ \( ( 1 - T )( 1 + T ) \)
$67$ \( ( 1 - T )( 1 + T ) \)
$71$ \( ( 1 - T )( 1 + T ) \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( ( 1 - T )( 1 + T ) \)
$83$ \( ( 1 - T )( 1 + T ) \)
$89$ \( 1 + T^{2} \)
$97$ \( ( 1 - T )( 1 + T ) \)
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