# Properties

 Label 1200.1.bj.a Level $1200$ Weight $1$ Character orbit 1200.bj Analytic conductor $0.599$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -3, -20, 60 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1200.bj (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-5})$$ Artin image: $OD_{16}:C_2$ Artin field: Galois closure of 16.0.2624400000000000000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{3} -2 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8} q^{3} -2 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} -2 q^{21} -\zeta_{8}^{3} q^{27} -2 \zeta_{8} q^{43} -3 \zeta_{8}^{2} q^{49} + 2 q^{61} + 2 \zeta_{8} q^{63} + 2 \zeta_{8}^{3} q^{67} - q^{81} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 8q^{21} + 8q^{61} - 4q^{81} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 −0.707107 + 0.707107i 0 0 0 1.41421 + 1.41421i 0 1.00000i 0
143.2 0 0.707107 0.707107i 0 0 0 −1.41421 1.41421i 0 1.00000i 0
1007.1 0 −0.707107 0.707107i 0 0 0 1.41421 1.41421i 0 1.00000i 0
1007.2 0 0.707107 + 0.707107i 0 0 0 −1.41421 + 1.41421i 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
60.h even 2 1 RM by $$\Q(\sqrt{15})$$
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
12.b even 2 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
20.e even 4 2 inner
60.l odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.1.bj.a 4
3.b odd 2 1 CM 1200.1.bj.a 4
4.b odd 2 1 inner 1200.1.bj.a 4
5.b even 2 1 inner 1200.1.bj.a 4
5.c odd 4 2 inner 1200.1.bj.a 4
12.b even 2 1 inner 1200.1.bj.a 4
15.d odd 2 1 inner 1200.1.bj.a 4
15.e even 4 2 inner 1200.1.bj.a 4
20.d odd 2 1 CM 1200.1.bj.a 4
20.e even 4 2 inner 1200.1.bj.a 4
60.h even 2 1 RM 1200.1.bj.a 4
60.l odd 4 2 inner 1200.1.bj.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.1.bj.a 4 1.a even 1 1 trivial
1200.1.bj.a 4 3.b odd 2 1 CM
1200.1.bj.a 4 4.b odd 2 1 inner
1200.1.bj.a 4 5.b even 2 1 inner
1200.1.bj.a 4 5.c odd 4 2 inner
1200.1.bj.a 4 12.b even 2 1 inner
1200.1.bj.a 4 15.d odd 2 1 inner
1200.1.bj.a 4 15.e even 4 2 inner
1200.1.bj.a 4 20.d odd 2 1 CM
1200.1.bj.a 4 20.e even 4 2 inner
1200.1.bj.a 4 60.h even 2 1 RM
1200.1.bj.a 4 60.l odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 16$$ acting on $$S_{1}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$16 + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$16 + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -2 + T )^{4}$$
$67$ $$16 + T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$