# Properties

 Label 1200.2.h.e 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: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 48) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 2 - 4 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 2 - 4 \zeta_{6} ) q^{7} -3 q^{9} + 2 q^{13} + ( 2 - 4 \zeta_{6} ) q^{19} + 6 q^{21} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 6 - 12 \zeta_{6} ) q^{31} + 10 q^{37} + ( -2 + 4 \zeta_{6} ) q^{39} + ( 6 - 12 \zeta_{6} ) q^{43} -5 q^{49} + 6 q^{57} + 14 q^{61} + ( -6 + 12 \zeta_{6} ) q^{63} + ( -2 + 4 \zeta_{6} ) q^{67} -10 q^{73} + ( -10 + 20 \zeta_{6} ) q^{79} + 9 q^{81} + ( 4 - 8 \zeta_{6} ) q^{91} + 18 q^{93} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{9} + O(q^{10})$$ $$2 q - 6 q^{9} + 4 q^{13} + 12 q^{21} + 20 q^{37} - 10 q^{49} + 12 q^{57} + 28 q^{61} - 20 q^{73} + 18 q^{81} + 36 q^{93} + 28 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0 0 3.46410i 0 −3.00000 0
1151.2 0 1.73205i 0 0 0 3.46410i 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.e 2
3.b odd 2 1 CM 1200.2.h.e 2
4.b odd 2 1 inner 1200.2.h.e 2
5.b even 2 1 48.2.c.a 2
5.c odd 4 2 1200.2.o.i 4
12.b even 2 1 inner 1200.2.h.e 2
15.d odd 2 1 48.2.c.a 2
15.e even 4 2 1200.2.o.i 4
20.d odd 2 1 48.2.c.a 2
20.e even 4 2 1200.2.o.i 4
35.c odd 2 1 2352.2.h.c 2
40.e odd 2 1 192.2.c.a 2
40.f even 2 1 192.2.c.a 2
45.h odd 6 1 1296.2.s.b 2
45.h odd 6 1 1296.2.s.e 2
45.j even 6 1 1296.2.s.b 2
45.j even 6 1 1296.2.s.e 2
60.h even 2 1 48.2.c.a 2
60.l odd 4 2 1200.2.o.i 4
80.k odd 4 2 768.2.f.d 4
80.q even 4 2 768.2.f.d 4
105.g even 2 1 2352.2.h.c 2
120.i odd 2 1 192.2.c.a 2
120.m even 2 1 192.2.c.a 2
140.c even 2 1 2352.2.h.c 2
180.n even 6 1 1296.2.s.b 2
180.n even 6 1 1296.2.s.e 2
180.p odd 6 1 1296.2.s.b 2
180.p odd 6 1 1296.2.s.e 2
240.t even 4 2 768.2.f.d 4
240.bm odd 4 2 768.2.f.d 4
420.o odd 2 1 2352.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.c.a 2 5.b even 2 1
48.2.c.a 2 15.d odd 2 1
48.2.c.a 2 20.d odd 2 1
48.2.c.a 2 60.h even 2 1
192.2.c.a 2 40.e odd 2 1
192.2.c.a 2 40.f even 2 1
192.2.c.a 2 120.i odd 2 1
192.2.c.a 2 120.m even 2 1
768.2.f.d 4 80.k odd 4 2
768.2.f.d 4 80.q even 4 2
768.2.f.d 4 240.t even 4 2
768.2.f.d 4 240.bm odd 4 2
1200.2.h.e 2 1.a even 1 1 trivial
1200.2.h.e 2 3.b odd 2 1 CM
1200.2.h.e 2 4.b odd 2 1 inner
1200.2.h.e 2 12.b even 2 1 inner
1200.2.o.i 4 5.c odd 4 2
1200.2.o.i 4 15.e even 4 2
1200.2.o.i 4 20.e even 4 2
1200.2.o.i 4 60.l odd 4 2
1296.2.s.b 2 45.h odd 6 1
1296.2.s.b 2 45.j even 6 1
1296.2.s.b 2 180.n even 6 1
1296.2.s.b 2 180.p odd 6 1
1296.2.s.e 2 45.h odd 6 1
1296.2.s.e 2 45.j even 6 1
1296.2.s.e 2 180.n even 6 1
1296.2.s.e 2 180.p odd 6 1
2352.2.h.c 2 35.c odd 2 1
2352.2.h.c 2 105.g even 2 1
2352.2.h.c 2 140.c even 2 1
2352.2.h.c 2 420.o odd 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 12$$ $$T_{11}$$ $$T_{13} - 2$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$12 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$12 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$108 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$108 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$12 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$300 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -14 + T )^{2}$$