# 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

## 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
 0.5 − 0.866025i 0.5 + 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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