# Properties

 Label 1200.2.h.m Level $1200$ Weight $2$ Character orbit 1200.h Analytic conductor $9.582$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{7} + 3 q^{9} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} -4 q^{13} + 6 \zeta_{12}^{3} q^{17} + ( 2 - 4 \zeta_{12}^{2} ) q^{19} + 6 \zeta_{12}^{3} q^{21} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{23} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} -6 \zeta_{12}^{3} q^{29} + ( -2 + 4 \zeta_{12}^{2} ) q^{31} + 6 q^{33} + 4 q^{37} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{39} + 12 \zeta_{12}^{3} q^{41} + ( -4 + 8 \zeta_{12}^{2} ) q^{43} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} -5 q^{49} + ( 6 - 12 \zeta_{12}^{2} ) q^{51} + 6 \zeta_{12}^{3} q^{53} + 6 \zeta_{12}^{3} q^{57} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} -10 q^{61} + ( 6 - 12 \zeta_{12}^{2} ) q^{63} + ( -4 + 8 \zeta_{12}^{2} ) q^{67} -6 q^{69} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{71} + 2 q^{73} + 12 \zeta_{12}^{3} q^{77} + ( -6 + 12 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{83} + ( -6 + 12 \zeta_{12}^{2} ) q^{87} + ( -8 + 16 \zeta_{12}^{2} ) q^{91} -6 \zeta_{12}^{3} q^{93} -10 q^{97} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} - 16q^{13} + 24q^{33} + 16q^{37} - 20q^{49} - 40q^{61} - 24q^{69} + 8q^{73} + 36q^{81} - 40q^{97} + O(q^{100})$$

## 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
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
0 −1.73205 0 0 0 3.46410i 0 3.00000 0
1151.2 0 −1.73205 0 0 0 3.46410i 0 3.00000 0
1151.3 0 1.73205 0 0 0 3.46410i 0 3.00000 0
1151.4 0 1.73205 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 inner
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.m 4
3.b odd 2 1 inner 1200.2.h.m 4
4.b odd 2 1 inner 1200.2.h.m 4
5.b even 2 1 240.2.h.b 4
5.c odd 4 1 1200.2.o.a 4
5.c odd 4 1 1200.2.o.b 4
12.b even 2 1 inner 1200.2.h.m 4
15.d odd 2 1 240.2.h.b 4
15.e even 4 1 1200.2.o.a 4
15.e even 4 1 1200.2.o.b 4
20.d odd 2 1 240.2.h.b 4
20.e even 4 1 1200.2.o.a 4
20.e even 4 1 1200.2.o.b 4
40.e odd 2 1 960.2.h.d 4
40.f even 2 1 960.2.h.d 4
60.h even 2 1 240.2.h.b 4
60.l odd 4 1 1200.2.o.a 4
60.l odd 4 1 1200.2.o.b 4
120.i odd 2 1 960.2.h.d 4
120.m even 2 1 960.2.h.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.h.b 4 5.b even 2 1
240.2.h.b 4 15.d odd 2 1
240.2.h.b 4 20.d odd 2 1
240.2.h.b 4 60.h even 2 1
960.2.h.d 4 40.e odd 2 1
960.2.h.d 4 40.f even 2 1
960.2.h.d 4 120.i odd 2 1
960.2.h.d 4 120.m even 2 1
1200.2.h.m 4 1.a even 1 1 trivial
1200.2.h.m 4 3.b odd 2 1 inner
1200.2.h.m 4 4.b odd 2 1 inner
1200.2.h.m 4 12.b even 2 1 inner
1200.2.o.a 4 5.c odd 4 1
1200.2.o.a 4 15.e even 4 1
1200.2.o.a 4 20.e even 4 1
1200.2.o.a 4 60.l odd 4 1
1200.2.o.b 4 5.c odd 4 1
1200.2.o.b 4 15.e even 4 1
1200.2.o.b 4 20.e even 4 1
1200.2.o.b 4 60.l odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 12$$ $$T_{11}^{2} - 12$$ $$T_{13} + 4$$ $$T_{23}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 12 + T^{2} )^{2}$$
$11$ $$( -12 + T^{2} )^{2}$$
$13$ $$( 4 + T )^{4}$$
$17$ $$( 36 + T^{2} )^{2}$$
$19$ $$( 12 + T^{2} )^{2}$$
$23$ $$( -12 + T^{2} )^{2}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$( 12 + T^{2} )^{2}$$
$37$ $$( -4 + T )^{4}$$
$41$ $$( 144 + T^{2} )^{2}$$
$43$ $$( 48 + T^{2} )^{2}$$
$47$ $$( -12 + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( -12 + T^{2} )^{2}$$
$61$ $$( 10 + T )^{4}$$
$67$ $$( 48 + T^{2} )^{2}$$
$71$ $$( -192 + T^{2} )^{2}$$
$73$ $$( -2 + T )^{4}$$
$79$ $$( 108 + T^{2} )^{2}$$
$83$ $$( -108 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( 10 + T )^{4}$$