Properties

Label 180.1.f.a
Level $180$
Weight $1$
Character orbit 180.f
Analytic conductor $0.090$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -15, 60
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 180.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0898317022739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{15})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.4665600.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} -i q^{5} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} -i q^{5} + i q^{8} - q^{10} + q^{16} + 2 i q^{17} + i q^{20} - q^{25} -i q^{32} + 2 q^{34} + q^{40} - q^{49} + i q^{50} -2 i q^{53} -2 q^{61} - q^{64} -2 i q^{68} -i q^{80} + 2 q^{85} + i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{10} + 2q^{16} - 2q^{25} + 4q^{34} + 2q^{40} - 2q^{49} - 4q^{61} - 2q^{64} + 4q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
1.00000i
1.00000i
1.00000i 0 −1.00000 1.00000i 0 0 1.00000i 0 −1.00000
19.2 1.00000i 0 −1.00000 1.00000i 0 0 1.00000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
60.h even 2 1 RM by \(\Q(\sqrt{15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.1.f.a 2
3.b odd 2 1 inner 180.1.f.a 2
4.b odd 2 1 CM 180.1.f.a 2
5.b even 2 1 inner 180.1.f.a 2
5.c odd 4 1 900.1.c.a 1
5.c odd 4 1 900.1.c.b 1
8.b even 2 1 2880.1.j.b 2
8.d odd 2 1 2880.1.j.b 2
9.c even 3 2 1620.1.p.b 4
9.d odd 6 2 1620.1.p.b 4
12.b even 2 1 inner 180.1.f.a 2
15.d odd 2 1 CM 180.1.f.a 2
15.e even 4 1 900.1.c.a 1
15.e even 4 1 900.1.c.b 1
20.d odd 2 1 inner 180.1.f.a 2
20.e even 4 1 900.1.c.a 1
20.e even 4 1 900.1.c.b 1
24.f even 2 1 2880.1.j.b 2
24.h odd 2 1 2880.1.j.b 2
36.f odd 6 2 1620.1.p.b 4
36.h even 6 2 1620.1.p.b 4
40.e odd 2 1 2880.1.j.b 2
40.f even 2 1 2880.1.j.b 2
45.h odd 6 2 1620.1.p.b 4
45.j even 6 2 1620.1.p.b 4
60.h even 2 1 RM 180.1.f.a 2
60.l odd 4 1 900.1.c.a 1
60.l odd 4 1 900.1.c.b 1
120.i odd 2 1 2880.1.j.b 2
120.m even 2 1 2880.1.j.b 2
180.n even 6 2 1620.1.p.b 4
180.p odd 6 2 1620.1.p.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.f.a 2 1.a even 1 1 trivial
180.1.f.a 2 3.b odd 2 1 inner
180.1.f.a 2 4.b odd 2 1 CM
180.1.f.a 2 5.b even 2 1 inner
180.1.f.a 2 12.b even 2 1 inner
180.1.f.a 2 15.d odd 2 1 CM
180.1.f.a 2 20.d odd 2 1 inner
180.1.f.a 2 60.h even 2 1 RM
900.1.c.a 1 5.c odd 4 1
900.1.c.a 1 15.e even 4 1
900.1.c.a 1 20.e even 4 1
900.1.c.a 1 60.l odd 4 1
900.1.c.b 1 5.c odd 4 1
900.1.c.b 1 15.e even 4 1
900.1.c.b 1 20.e even 4 1
900.1.c.b 1 60.l odd 4 1
1620.1.p.b 4 9.c even 3 2
1620.1.p.b 4 9.d odd 6 2
1620.1.p.b 4 36.f odd 6 2
1620.1.p.b 4 36.h even 6 2
1620.1.p.b 4 45.h odd 6 2
1620.1.p.b 4 45.j even 6 2
1620.1.p.b 4 180.n even 6 2
1620.1.p.b 4 180.p odd 6 2
2880.1.j.b 2 8.b even 2 1
2880.1.j.b 2 8.d odd 2 1
2880.1.j.b 2 24.f even 2 1
2880.1.j.b 2 24.h odd 2 1
2880.1.j.b 2 40.e odd 2 1
2880.1.j.b 2 40.f even 2 1
2880.1.j.b 2 120.i odd 2 1
2880.1.j.b 2 120.m even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(180, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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