# Properties

 Label 240.1.bm.a Level $240$ Weight $1$ Character orbit 240.bm Analytic conductor $0.120$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 240.bm (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.119775603032$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.92160.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{2} + \zeta_{8} q^{3} + \zeta_{8}^{2} q^{4} + \zeta_{8}^{3} q^{5} -\zeta_{8}^{2} q^{6} -\zeta_{8}^{3} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8} q^{2} + \zeta_{8} q^{3} + \zeta_{8}^{2} q^{4} + \zeta_{8}^{3} q^{5} -\zeta_{8}^{2} q^{6} -\zeta_{8}^{3} q^{8} + \zeta_{8}^{2} q^{9} + q^{10} + \zeta_{8}^{3} q^{12} - q^{15} - q^{16} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{17} -\zeta_{8}^{3} q^{18} + ( -1 - \zeta_{8}^{2} ) q^{19} -\zeta_{8} q^{20} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{23} + q^{24} -\zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + \zeta_{8} q^{30} + \zeta_{8} q^{32} + ( -1 - \zeta_{8}^{2} ) q^{34} - q^{36} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{38} + \zeta_{8}^{2} q^{40} -\zeta_{8} q^{45} + ( -1 + \zeta_{8}^{2} ) q^{46} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{47} -\zeta_{8} q^{48} - q^{49} + \zeta_{8}^{3} q^{50} + ( 1 + \zeta_{8}^{2} ) q^{51} + q^{54} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{57} -\zeta_{8}^{2} q^{60} + ( 1 + \zeta_{8}^{2} ) q^{61} -\zeta_{8}^{2} q^{64} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{68} + ( 1 - \zeta_{8}^{2} ) q^{69} + \zeta_{8} q^{72} -\zeta_{8}^{3} q^{75} + ( 1 - \zeta_{8}^{2} ) q^{76} + 2 q^{79} -\zeta_{8}^{3} q^{80} - q^{81} + ( -1 + \zeta_{8}^{2} ) q^{85} + \zeta_{8}^{2} q^{90} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{92} + ( 1 + \zeta_{8}^{2} ) q^{94} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{95} + \zeta_{8}^{2} q^{96} + \zeta_{8} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 4 q^{10} - 4 q^{15} - 4 q^{16} - 4 q^{19} + 4 q^{24} - 4 q^{34} - 4 q^{36} - 4 q^{46} - 4 q^{49} + 4 q^{51} + 4 q^{54} + 4 q^{61} + 4 q^{69} + 4 q^{76} + 8 q^{79} - 4 q^{81} - 4 q^{85} + 4 q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
−0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −0.707107 + 0.707107i 1.00000i 0 0.707107 0.707107i 1.00000i 1.00000
29.2 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0.707107 0.707107i 1.00000i 0 −0.707107 + 0.707107i 1.00000i 1.00000
149.1 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −0.707107 0.707107i 1.00000i 0 0.707107 + 0.707107i 1.00000i 1.00000
149.2 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0.707107 + 0.707107i 1.00000i 0 −0.707107 0.707107i 1.00000i 1.00000
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner
80.q even 4 1 inner
240.bm odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.1.bm.a 4
3.b odd 2 1 inner 240.1.bm.a 4
4.b odd 2 1 960.1.bm.a 4
5.b even 2 1 inner 240.1.bm.a 4
5.c odd 4 2 1200.1.r.a 4
8.b even 2 1 1920.1.bm.a 4
8.d odd 2 1 1920.1.bm.b 4
12.b even 2 1 960.1.bm.a 4
15.d odd 2 1 CM 240.1.bm.a 4
15.e even 4 2 1200.1.r.a 4
16.e even 4 1 inner 240.1.bm.a 4
16.e even 4 1 1920.1.bm.a 4
16.f odd 4 1 960.1.bm.a 4
16.f odd 4 1 1920.1.bm.b 4
20.d odd 2 1 960.1.bm.a 4
24.f even 2 1 1920.1.bm.b 4
24.h odd 2 1 1920.1.bm.a 4
40.e odd 2 1 1920.1.bm.b 4
40.f even 2 1 1920.1.bm.a 4
48.i odd 4 1 inner 240.1.bm.a 4
48.i odd 4 1 1920.1.bm.a 4
48.k even 4 1 960.1.bm.a 4
48.k even 4 1 1920.1.bm.b 4
60.h even 2 1 960.1.bm.a 4
80.i odd 4 1 1200.1.r.a 4
80.k odd 4 1 960.1.bm.a 4
80.k odd 4 1 1920.1.bm.b 4
80.q even 4 1 inner 240.1.bm.a 4
80.q even 4 1 1920.1.bm.a 4
80.t odd 4 1 1200.1.r.a 4
120.i odd 2 1 1920.1.bm.a 4
120.m even 2 1 1920.1.bm.b 4
240.t even 4 1 960.1.bm.a 4
240.t even 4 1 1920.1.bm.b 4
240.bb even 4 1 1200.1.r.a 4
240.bf even 4 1 1200.1.r.a 4
240.bm odd 4 1 inner 240.1.bm.a 4
240.bm odd 4 1 1920.1.bm.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.1.bm.a 4 1.a even 1 1 trivial
240.1.bm.a 4 3.b odd 2 1 inner
240.1.bm.a 4 5.b even 2 1 inner
240.1.bm.a 4 15.d odd 2 1 CM
240.1.bm.a 4 16.e even 4 1 inner
240.1.bm.a 4 48.i odd 4 1 inner
240.1.bm.a 4 80.q even 4 1 inner
240.1.bm.a 4 240.bm odd 4 1 inner
960.1.bm.a 4 4.b odd 2 1
960.1.bm.a 4 12.b even 2 1
960.1.bm.a 4 16.f odd 4 1
960.1.bm.a 4 20.d odd 2 1
960.1.bm.a 4 48.k even 4 1
960.1.bm.a 4 60.h even 2 1
960.1.bm.a 4 80.k odd 4 1
960.1.bm.a 4 240.t even 4 1
1200.1.r.a 4 5.c odd 4 2
1200.1.r.a 4 15.e even 4 2
1200.1.r.a 4 80.i odd 4 1
1200.1.r.a 4 80.t odd 4 1
1200.1.r.a 4 240.bb even 4 1
1200.1.r.a 4 240.bf even 4 1
1920.1.bm.a 4 8.b even 2 1
1920.1.bm.a 4 16.e even 4 1
1920.1.bm.a 4 24.h odd 2 1
1920.1.bm.a 4 40.f even 2 1
1920.1.bm.a 4 48.i odd 4 1
1920.1.bm.a 4 80.q even 4 1
1920.1.bm.a 4 120.i odd 2 1
1920.1.bm.a 4 240.bm odd 4 1
1920.1.bm.b 4 8.d odd 2 1
1920.1.bm.b 4 16.f odd 4 1
1920.1.bm.b 4 24.f even 2 1
1920.1.bm.b 4 40.e odd 2 1
1920.1.bm.b 4 48.k even 4 1
1920.1.bm.b 4 80.k odd 4 1
1920.1.bm.b 4 120.m even 2 1
1920.1.bm.b 4 240.t even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(240, [\chi])$$.

## Hecke characteristic polynomials

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