Properties

Label 180.1.p.b
Level $180$
Weight $1$
Character orbit 180.p
Analytic conductor $0.090$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -20
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 180.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0898317022739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1620.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.419904000000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{5} + q^{6} + \zeta_{6}^{2} q^{7} - q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{5} + q^{6} + \zeta_{6}^{2} q^{7} - q^{8} + \zeta_{6}^{2} q^{9} - q^{10} -\zeta_{6}^{2} q^{12} + \zeta_{6} q^{14} -\zeta_{6}^{2} q^{15} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} + \zeta_{6}^{2} q^{20} - q^{21} -\zeta_{6} q^{23} -\zeta_{6} q^{24} + \zeta_{6}^{2} q^{25} - q^{27} + q^{28} -\zeta_{6}^{2} q^{29} -\zeta_{6} q^{30} + \zeta_{6} q^{32} + q^{35} + q^{36} + \zeta_{6} q^{40} + \zeta_{6} q^{41} + \zeta_{6}^{2} q^{42} -2 \zeta_{6}^{2} q^{43} + q^{45} - q^{46} + \zeta_{6}^{2} q^{47} - q^{48} + \zeta_{6} q^{50} + \zeta_{6}^{2} q^{54} -\zeta_{6}^{2} q^{56} -\zeta_{6} q^{58} - q^{60} -\zeta_{6}^{2} q^{61} -\zeta_{6} q^{63} + q^{64} -\zeta_{6} q^{67} -\zeta_{6}^{2} q^{69} -\zeta_{6}^{2} q^{70} -\zeta_{6}^{2} q^{72} - q^{75} + q^{80} -\zeta_{6} q^{81} + q^{82} + \zeta_{6}^{2} q^{83} + \zeta_{6} q^{84} -2 \zeta_{6} q^{86} + q^{87} - q^{89} -\zeta_{6}^{2} q^{90} + \zeta_{6}^{2} q^{92} + \zeta_{6} q^{94} + \zeta_{6}^{2} q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{3} - q^{4} - q^{5} + 2q^{6} - q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} + q^{3} - q^{4} - q^{5} + 2q^{6} - q^{7} - 2q^{8} - q^{9} - 2q^{10} + q^{12} + q^{14} + q^{15} - q^{16} + q^{18} - q^{20} - 2q^{21} - q^{23} - q^{24} - q^{25} - 2q^{27} + 2q^{28} + q^{29} - q^{30} + q^{32} + 2q^{35} + 2q^{36} + q^{40} + q^{41} - q^{42} + 2q^{43} + 2q^{45} - 2q^{46} - q^{47} - 2q^{48} + q^{50} - q^{54} + q^{56} - q^{58} - 2q^{60} + q^{61} - q^{63} + 2q^{64} - q^{67} + q^{69} + q^{70} + q^{72} - 2q^{75} + 2q^{80} - q^{81} + 2q^{82} - q^{83} + q^{84} - 2q^{86} + 2q^{87} - 2q^{89} + q^{90} - q^{92} + q^{94} - q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i −1.00000
139.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −1.00000 −0.500000 0.866025i −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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
9.c even 3 1 inner
180.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.1.p.b yes 2
3.b odd 2 1 540.1.p.a 2
4.b odd 2 1 180.1.p.a 2
5.b even 2 1 180.1.p.a 2
5.c odd 4 2 900.1.t.a 4
8.b even 2 1 2880.1.bu.a 2
8.d odd 2 1 2880.1.bu.b 2
9.c even 3 1 inner 180.1.p.b yes 2
9.c even 3 1 1620.1.f.b 1
9.d odd 6 1 540.1.p.a 2
9.d odd 6 1 1620.1.f.c 1
12.b even 2 1 540.1.p.b 2
15.d odd 2 1 540.1.p.b 2
15.e even 4 2 2700.1.t.a 4
20.d odd 2 1 CM 180.1.p.b yes 2
20.e even 4 2 900.1.t.a 4
36.f odd 6 1 180.1.p.a 2
36.f odd 6 1 1620.1.f.d 1
36.h even 6 1 540.1.p.b 2
36.h even 6 1 1620.1.f.a 1
40.e odd 2 1 2880.1.bu.a 2
40.f even 2 1 2880.1.bu.b 2
45.h odd 6 1 540.1.p.b 2
45.h odd 6 1 1620.1.f.a 1
45.j even 6 1 180.1.p.a 2
45.j even 6 1 1620.1.f.d 1
45.k odd 12 2 900.1.t.a 4
45.l even 12 2 2700.1.t.a 4
60.h even 2 1 540.1.p.a 2
60.l odd 4 2 2700.1.t.a 4
72.n even 6 1 2880.1.bu.a 2
72.p odd 6 1 2880.1.bu.b 2
180.n even 6 1 540.1.p.a 2
180.n even 6 1 1620.1.f.c 1
180.p odd 6 1 inner 180.1.p.b yes 2
180.p odd 6 1 1620.1.f.b 1
180.v odd 12 2 2700.1.t.a 4
180.x even 12 2 900.1.t.a 4
360.z odd 6 1 2880.1.bu.a 2
360.bk even 6 1 2880.1.bu.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.p.a 2 4.b odd 2 1
180.1.p.a 2 5.b even 2 1
180.1.p.a 2 36.f odd 6 1
180.1.p.a 2 45.j even 6 1
180.1.p.b yes 2 1.a even 1 1 trivial
180.1.p.b yes 2 9.c even 3 1 inner
180.1.p.b yes 2 20.d odd 2 1 CM
180.1.p.b yes 2 180.p odd 6 1 inner
540.1.p.a 2 3.b odd 2 1
540.1.p.a 2 9.d odd 6 1
540.1.p.a 2 60.h even 2 1
540.1.p.a 2 180.n even 6 1
540.1.p.b 2 12.b even 2 1
540.1.p.b 2 15.d odd 2 1
540.1.p.b 2 36.h even 6 1
540.1.p.b 2 45.h odd 6 1
900.1.t.a 4 5.c odd 4 2
900.1.t.a 4 20.e even 4 2
900.1.t.a 4 45.k odd 12 2
900.1.t.a 4 180.x even 12 2
1620.1.f.a 1 36.h even 6 1
1620.1.f.a 1 45.h odd 6 1
1620.1.f.b 1 9.c even 3 1
1620.1.f.b 1 180.p odd 6 1
1620.1.f.c 1 9.d odd 6 1
1620.1.f.c 1 180.n even 6 1
1620.1.f.d 1 36.f odd 6 1
1620.1.f.d 1 45.j even 6 1
2700.1.t.a 4 15.e even 4 2
2700.1.t.a 4 45.l even 12 2
2700.1.t.a 4 60.l odd 4 2
2700.1.t.a 4 180.v odd 12 2
2880.1.bu.a 2 8.b even 2 1
2880.1.bu.a 2 40.e odd 2 1
2880.1.bu.a 2 72.n even 6 1
2880.1.bu.a 2 360.z odd 6 1
2880.1.bu.b 2 8.d odd 2 1
2880.1.bu.b 2 40.f even 2 1
2880.1.bu.b 2 72.p odd 6 1
2880.1.bu.b 2 360.bk even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(180, [\chi])\).

Hecke characteristic polynomials

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