# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - 3 \beta q^{7} - 3 q^{9} +O(q^{10})$$ q - b * q^3 - 3*b * q^7 - 3 * q^9 $$q - \beta q^{3} - 3 \beta q^{7} - 3 q^{9} + 7 q^{13} - 3 \beta q^{19} - 9 q^{21} + 3 \beta q^{27} + \beta q^{31} - 10 q^{37} - 7 \beta q^{39} + \beta q^{43} - 20 q^{49} - 9 q^{57} - q^{61} + 9 \beta q^{63} - 7 \beta q^{67} + 10 q^{73} + 10 \beta q^{79} + 9 q^{81} - 21 \beta q^{91} + 3 q^{93} + 19 q^{97} +O(q^{100})$$ q - b * q^3 - 3*b * q^7 - 3 * q^9 + 7 * q^13 - 3*b * q^19 - 9 * q^21 + 3*b * q^27 + b * q^31 - 10 * q^37 - 7*b * q^39 + b * q^43 - 20 * q^49 - 9 * q^57 - q^61 + 9*b * q^63 - 7*b * q^67 + 10 * q^73 + 10*b * q^79 + 9 * q^81 - 21*b * q^91 + 3 * q^93 + 19 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{9}+O(q^{10})$$ 2 * q - 6 * q^9 $$2 q - 6 q^{9} + 14 q^{13} - 18 q^{21} - 20 q^{37} - 40 q^{49} - 18 q^{57} - 2 q^{61} + 20 q^{73} + 18 q^{81} + 6 q^{93} + 38 q^{97}+O(q^{100})$$ 2 * q - 6 * q^9 + 14 * q^13 - 18 * q^21 - 20 * q^37 - 40 * q^49 - 18 * q^57 - 2 * q^61 + 20 * q^73 + 18 * q^81 + 6 * q^93 + 38 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 0 0 5.19615i 0 −3.00000 0
1151.2 0 1.73205i 0 0 0 5.19615i 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
12.b 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.h.g yes 2
3.b odd 2 1 CM 1200.2.h.g yes 2
4.b odd 2 1 inner 1200.2.h.g yes 2
5.b even 2 1 1200.2.h.c 2
5.c odd 4 2 1200.2.o.g 4
12.b even 2 1 inner 1200.2.h.g yes 2
15.d odd 2 1 1200.2.h.c 2
15.e even 4 2 1200.2.o.g 4
20.d odd 2 1 1200.2.h.c 2
20.e even 4 2 1200.2.o.g 4
60.h even 2 1 1200.2.h.c 2
60.l odd 4 2 1200.2.o.g 4

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

## Hecke kernels

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

 $$T_{7}^{2} + 27$$ T7^2 + 27 $$T_{11}$$ T11 $$T_{13} - 7$$ T13 - 7 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 27$$
$11$ $$T^{2}$$
$13$ $$(T - 7)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 27$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 3$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 3$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2} + 147$$
$71$ $$T^{2}$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} + 300$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 19)^{2}$$