# Properties

 Label 1200.2.o.g Level $1200$ Weight $2$ Character orbit 1200.o Analytic conductor $9.582$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + 7 \zeta_{12}^{3} q^{13} + ( 3 - 6 \zeta_{12}^{2} ) q^{19} -9 q^{21} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 1 - 2 \zeta_{12}^{2} ) q^{31} + 10 \zeta_{12}^{3} q^{37} + ( 7 - 14 \zeta_{12}^{2} ) q^{39} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{43} + 20 q^{49} + 9 \zeta_{12}^{3} q^{57} - q^{61} + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{63} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{67} + 10 \zeta_{12}^{3} q^{73} + ( -10 + 20 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( -21 + 42 \zeta_{12}^{2} ) q^{91} + 3 \zeta_{12}^{3} q^{93} -19 \zeta_{12}^{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} - 36q^{21} + 80q^{49} - 4q^{61} + 36q^{81} + O(q^{100})$$

## Character values

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

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1199.1
 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
0 −1.73205 0 0 0 5.19615 0 3.00000 0
1199.2 0 −1.73205 0 0 0 5.19615 0 3.00000 0
1199.3 0 1.73205 0 0 0 −5.19615 0 3.00000 0
1199.4 0 1.73205 0 0 0 −5.19615 0 3.00000 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 $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.o.g 4
3.b odd 2 1 CM 1200.2.o.g 4
4.b odd 2 1 inner 1200.2.o.g 4
5.b even 2 1 inner 1200.2.o.g 4
5.c odd 4 1 1200.2.h.c 2
5.c odd 4 1 1200.2.h.g yes 2
12.b even 2 1 inner 1200.2.o.g 4
15.d odd 2 1 inner 1200.2.o.g 4
15.e even 4 1 1200.2.h.c 2
15.e even 4 1 1200.2.h.g yes 2
20.d odd 2 1 inner 1200.2.o.g 4
20.e even 4 1 1200.2.h.c 2
20.e even 4 1 1200.2.h.g yes 2
60.h even 2 1 inner 1200.2.o.g 4
60.l odd 4 1 1200.2.h.c 2
60.l odd 4 1 1200.2.h.g yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.2.h.c 2 5.c odd 4 1
1200.2.h.c 2 15.e even 4 1
1200.2.h.c 2 20.e even 4 1
1200.2.h.c 2 60.l odd 4 1
1200.2.h.g yes 2 5.c odd 4 1
1200.2.h.g yes 2 15.e even 4 1
1200.2.h.g yes 2 20.e even 4 1
1200.2.h.g yes 2 60.l odd 4 1
1200.2.o.g 4 1.a even 1 1 trivial
1200.2.o.g 4 3.b odd 2 1 CM
1200.2.o.g 4 4.b odd 2 1 inner
1200.2.o.g 4 5.b even 2 1 inner
1200.2.o.g 4 12.b even 2 1 inner
1200.2.o.g 4 15.d odd 2 1 inner
1200.2.o.g 4 20.d odd 2 1 inner
1200.2.o.g 4 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} - 27$$ $$T_{11}$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -27 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 49 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 27 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$( 100 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -3 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 1 + T )^{4}$$
$67$ $$( -147 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 100 + T^{2} )^{2}$$
$79$ $$( 300 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$( 361 + T^{2} )^{2}$$