# Properties

 Label 1200.1.r.a Level $1200$ Weight $1$ Character orbit 1200.r Analytic conductor $0.599$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1200.r (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.598878015160$$ 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: no (minimal twist has level 240) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.92160.2

## $q$-expansion

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 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 1.00000i 0 0.707107 0.707107i 1.00000i 0
101.2 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i 0 −0.707107 + 0.707107i 1.00000i 0
701.1 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i 0 0.707107 + 0.707107i 1.00000i 0
701.2 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i 0 −0.707107 0.707107i 1.00000i 0
 $$n$$: 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 1200.1.r.a 4
3.b odd 2 1 inner 1200.1.r.a 4
5.b even 2 1 inner 1200.1.r.a 4
5.c odd 4 2 240.1.bm.a 4
15.d odd 2 1 CM 1200.1.r.a 4
15.e even 4 2 240.1.bm.a 4
16.e even 4 1 inner 1200.1.r.a 4
20.e even 4 2 960.1.bm.a 4
40.i odd 4 2 1920.1.bm.a 4
40.k even 4 2 1920.1.bm.b 4
48.i odd 4 1 inner 1200.1.r.a 4
60.l odd 4 2 960.1.bm.a 4
80.i odd 4 1 240.1.bm.a 4
80.i odd 4 1 1920.1.bm.a 4
80.j even 4 1 960.1.bm.a 4
80.j even 4 1 1920.1.bm.b 4
80.q even 4 1 inner 1200.1.r.a 4
80.s even 4 1 960.1.bm.a 4
80.s even 4 1 1920.1.bm.b 4
80.t odd 4 1 240.1.bm.a 4
80.t odd 4 1 1920.1.bm.a 4
120.q odd 4 2 1920.1.bm.b 4
120.w even 4 2 1920.1.bm.a 4
240.z odd 4 1 960.1.bm.a 4
240.z odd 4 1 1920.1.bm.b 4
240.bb even 4 1 240.1.bm.a 4
240.bb even 4 1 1920.1.bm.a 4
240.bd odd 4 1 960.1.bm.a 4
240.bd odd 4 1 1920.1.bm.b 4
240.bf even 4 1 240.1.bm.a 4
240.bf even 4 1 1920.1.bm.a 4
240.bm odd 4 1 inner 1200.1.r.a 4

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

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