Properties

Label 155.1.c.b
Level 155
Weight 1
Character orbit 155.c
Analytic conductor 0.077
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM discriminant -31
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 155.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0773550769581\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.120125.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{2} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{4} + \zeta_{6} q^{5} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{7} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{2} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{4} + \zeta_{6} q^{5} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{7} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{8} - q^{9} + ( -1 + \zeta_{6}^{2} ) q^{10} + ( 2 + \zeta_{6} - \zeta_{6}^{2} ) q^{14} + q^{16} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{18} + q^{19} + ( -1 - \zeta_{6} - \zeta_{6}^{2} ) q^{20} + \zeta_{6}^{2} q^{25} + ( 2 \zeta_{6} + 2 \zeta_{6}^{2} ) q^{28} - q^{31} + ( 1 - \zeta_{6}^{2} ) q^{35} + ( 1 + \zeta_{6} - \zeta_{6}^{2} ) q^{36} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{38} + ( 1 - \zeta_{6}^{2} ) q^{40} - q^{41} -\zeta_{6} q^{45} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{49} + ( -1 - \zeta_{6} ) q^{50} + ( -2 - \zeta_{6} + \zeta_{6}^{2} ) q^{56} - q^{59} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{62} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{63} + q^{64} + ( 1 + 2 \zeta_{6} + \zeta_{6}^{2} ) q^{70} + q^{71} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{72} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{76} + \zeta_{6} q^{80} + q^{81} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{82} + ( 1 - \zeta_{6}^{2} ) q^{90} + \zeta_{6} q^{95} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{97} + ( -2 \zeta_{6} - 2 \zeta_{6}^{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + q^{5} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{4} + q^{5} - 2q^{9} - 3q^{10} + 6q^{14} + 2q^{16} + 2q^{19} - 2q^{20} - q^{25} - 2q^{31} + 3q^{35} + 4q^{36} + 3q^{40} - 2q^{41} - q^{45} - 4q^{49} - 3q^{50} - 6q^{56} - 2q^{59} + 2q^{64} + 3q^{70} + 2q^{71} - 4q^{76} + q^{80} + 2q^{81} + 3q^{90} + q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(96\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
154.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i 0 −2.00000 0.500000 0.866025i 0 1.73205i 1.73205i −1.00000 −1.50000 0.866025i
154.2 1.73205i 0 −2.00000 0.500000 + 0.866025i 0 1.73205i 1.73205i −1.00000 −1.50000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
5.b even 2 1 inner
155.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.1.c.b 2
3.b odd 2 1 1395.1.b.c 2
4.b odd 2 1 2480.1.k.e 2
5.b even 2 1 inner 155.1.c.b 2
5.c odd 4 2 775.1.d.c 2
15.d odd 2 1 1395.1.b.c 2
20.d odd 2 1 2480.1.k.e 2
31.b odd 2 1 CM 155.1.c.b 2
93.c even 2 1 1395.1.b.c 2
124.d even 2 1 2480.1.k.e 2
155.c odd 2 1 inner 155.1.c.b 2
155.f even 4 2 775.1.d.c 2
465.g even 2 1 1395.1.b.c 2
620.e even 2 1 2480.1.k.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.1.c.b 2 1.a even 1 1 trivial
155.1.c.b 2 5.b even 2 1 inner
155.1.c.b 2 31.b odd 2 1 CM
155.1.c.b 2 155.c odd 2 1 inner
775.1.d.c 2 5.c odd 4 2
775.1.d.c 2 155.f even 4 2
1395.1.b.c 2 3.b odd 2 1
1395.1.b.c 2 15.d odd 2 1
1395.1.b.c 2 93.c even 2 1
1395.1.b.c 2 465.g even 2 1
2480.1.k.e 2 4.b odd 2 1
2480.1.k.e 2 20.d odd 2 1
2480.1.k.e 2 124.d even 2 1
2480.1.k.e 2 620.e even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(155, [\chi])\).

Hecke characteristic polynomials

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