Properties

Label 156.1.s.a
Level $156$
Weight $1$
Character orbit 156.s
Analytic conductor $0.078$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 156.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0778541419707\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.160398576.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} + ( -1 - \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{3} + ( -1 - \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} -\zeta_{6}^{2} q^{13} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{21} - q^{25} - q^{27} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{31} + q^{39} -\zeta_{6}^{2} q^{43} + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{49} + \zeta_{6}^{2} q^{61} + ( 1 - \zeta_{6}^{2} ) q^{63} + ( 1 - \zeta_{6}^{2} ) q^{67} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{73} -\zeta_{6} q^{75} + q^{79} -\zeta_{6} q^{81} + ( -1 + \zeta_{6}^{2} ) q^{91} + ( -1 + \zeta_{6}^{2} ) q^{93} + ( -1 - \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 3q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - 3q^{7} - q^{9} + q^{13} - 2q^{25} - 2q^{27} + 2q^{39} + q^{43} + 2q^{49} - q^{61} + 3q^{63} + 3q^{67} - q^{75} + 2q^{79} - q^{81} - 3q^{91} - 3q^{93} - 3q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 0.866025i 0 0 0 −1.50000 + 0.866025i 0 −0.500000 0.866025i 0
101.1 0 0.500000 + 0.866025i 0 0 0 −1.50000 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): 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 CM by \(\Q(\sqrt{-3}) \)
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.1.s.a 2
3.b odd 2 1 CM 156.1.s.a 2
4.b odd 2 1 624.1.cb.a 2
5.b even 2 1 3900.1.ca.b 2
5.c odd 4 2 3900.1.br.b 4
8.b even 2 1 2496.1.cb.a 2
8.d odd 2 1 2496.1.cb.b 2
12.b even 2 1 624.1.cb.a 2
13.b even 2 1 2028.1.s.a 2
13.c even 3 1 2028.1.g.a 2
13.c even 3 1 2028.1.s.a 2
13.d odd 4 2 2028.1.o.b 4
13.e even 6 1 inner 156.1.s.a 2
13.e even 6 1 2028.1.g.a 2
13.f odd 12 2 2028.1.d.c 2
13.f odd 12 2 2028.1.o.b 4
15.d odd 2 1 3900.1.ca.b 2
15.e even 4 2 3900.1.br.b 4
24.f even 2 1 2496.1.cb.b 2
24.h odd 2 1 2496.1.cb.a 2
39.d odd 2 1 2028.1.s.a 2
39.f even 4 2 2028.1.o.b 4
39.h odd 6 1 inner 156.1.s.a 2
39.h odd 6 1 2028.1.g.a 2
39.i odd 6 1 2028.1.g.a 2
39.i odd 6 1 2028.1.s.a 2
39.k even 12 2 2028.1.d.c 2
39.k even 12 2 2028.1.o.b 4
52.i odd 6 1 624.1.cb.a 2
65.l even 6 1 3900.1.ca.b 2
65.r odd 12 2 3900.1.br.b 4
104.p odd 6 1 2496.1.cb.b 2
104.s even 6 1 2496.1.cb.a 2
156.r even 6 1 624.1.cb.a 2
195.y odd 6 1 3900.1.ca.b 2
195.bf even 12 2 3900.1.br.b 4
312.ba even 6 1 2496.1.cb.b 2
312.bg odd 6 1 2496.1.cb.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.1.s.a 2 1.a even 1 1 trivial
156.1.s.a 2 3.b odd 2 1 CM
156.1.s.a 2 13.e even 6 1 inner
156.1.s.a 2 39.h odd 6 1 inner
624.1.cb.a 2 4.b odd 2 1
624.1.cb.a 2 12.b even 2 1
624.1.cb.a 2 52.i odd 6 1
624.1.cb.a 2 156.r even 6 1
2028.1.d.c 2 13.f odd 12 2
2028.1.d.c 2 39.k even 12 2
2028.1.g.a 2 13.c even 3 1
2028.1.g.a 2 13.e even 6 1
2028.1.g.a 2 39.h odd 6 1
2028.1.g.a 2 39.i odd 6 1
2028.1.o.b 4 13.d odd 4 2
2028.1.o.b 4 13.f odd 12 2
2028.1.o.b 4 39.f even 4 2
2028.1.o.b 4 39.k even 12 2
2028.1.s.a 2 13.b even 2 1
2028.1.s.a 2 13.c even 3 1
2028.1.s.a 2 39.d odd 2 1
2028.1.s.a 2 39.i odd 6 1
2496.1.cb.a 2 8.b even 2 1
2496.1.cb.a 2 24.h odd 2 1
2496.1.cb.a 2 104.s even 6 1
2496.1.cb.a 2 312.bg odd 6 1
2496.1.cb.b 2 8.d odd 2 1
2496.1.cb.b 2 24.f even 2 1
2496.1.cb.b 2 104.p odd 6 1
2496.1.cb.b 2 312.ba even 6 1
3900.1.br.b 4 5.c odd 4 2
3900.1.br.b 4 15.e even 4 2
3900.1.br.b 4 65.r odd 12 2
3900.1.br.b 4 195.bf even 12 2
3900.1.ca.b 2 5.b even 2 1
3900.1.ca.b 2 15.d odd 2 1
3900.1.ca.b 2 65.l even 6 1
3900.1.ca.b 2 195.y odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(156, [\chi])\).

Hecke characteristic polynomials

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