# Properties

 Label 1200.2.h.j 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{3} + ( -1 - 2 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{3} + ( -1 - 2 \beta_{1} ) q^{9} + \beta_{3} q^{11} -\beta_{3} q^{13} + \beta_{2} q^{17} + \beta_{2} q^{19} + 6 q^{23} + ( 5 + \beta_{1} ) q^{27} + 2 \beta_{1} q^{29} + \beta_{2} q^{31} + ( 2 \beta_{2} - \beta_{3} ) q^{33} -\beta_{3} q^{37} + ( -2 \beta_{2} + \beta_{3} ) q^{39} -4 \beta_{1} q^{41} + 6 \beta_{1} q^{43} -6 q^{47} + 7 q^{49} + ( -\beta_{2} - \beta_{3} ) q^{51} + 3 \beta_{2} q^{53} + ( -\beta_{2} - \beta_{3} ) q^{57} -\beta_{3} q^{59} -2 q^{61} + 6 \beta_{1} q^{67} + ( -6 + 6 \beta_{1} ) q^{69} -2 \beta_{3} q^{71} -2 \beta_{3} q^{73} + 3 \beta_{2} q^{79} + ( -7 + 4 \beta_{1} ) q^{81} + 6 q^{83} + ( -4 - 2 \beta_{1} ) q^{87} -4 \beta_{1} q^{89} + ( -\beta_{2} - \beta_{3} ) q^{93} + ( -4 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} - 4 q^{9} + 24 q^{23} + 20 q^{27} - 24 q^{47} + 28 q^{49} - 8 q^{61} - 24 q^{69} - 28 q^{81} + 24 q^{83} - 16 q^{87} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## 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
 −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i
0 −1.00000 1.41421i 0 0 0 0 0 −1.00000 + 2.82843i 0
1151.2 0 −1.00000 1.41421i 0 0 0 0 0 −1.00000 + 2.82843i 0
1151.3 0 −1.00000 + 1.41421i 0 0 0 0 0 −1.00000 2.82843i 0
1151.4 0 −1.00000 + 1.41421i 0 0 0 0 0 −1.00000 2.82843i 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
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 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.j 4
3.b odd 2 1 1200.2.h.n 4
4.b odd 2 1 1200.2.h.n 4
5.b even 2 1 1200.2.h.n 4
5.c odd 4 2 240.2.o.b 8
12.b even 2 1 inner 1200.2.h.j 4
15.d odd 2 1 inner 1200.2.h.j 4
15.e even 4 2 240.2.o.b 8
20.d odd 2 1 inner 1200.2.h.j 4
20.e even 4 2 240.2.o.b 8
40.i odd 4 2 960.2.o.d 8
40.k even 4 2 960.2.o.d 8
60.h even 2 1 1200.2.h.n 4
60.l odd 4 2 240.2.o.b 8
120.q odd 4 2 960.2.o.d 8
120.w even 4 2 960.2.o.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.o.b 8 5.c odd 4 2
240.2.o.b 8 15.e even 4 2
240.2.o.b 8 20.e even 4 2
240.2.o.b 8 60.l odd 4 2
960.2.o.d 8 40.i odd 4 2
960.2.o.d 8 40.k even 4 2
960.2.o.d 8 120.q odd 4 2
960.2.o.d 8 120.w even 4 2
1200.2.h.j 4 1.a even 1 1 trivial
1200.2.h.j 4 12.b even 2 1 inner
1200.2.h.j 4 15.d odd 2 1 inner
1200.2.h.j 4 20.d odd 2 1 inner
1200.2.h.n 4 3.b odd 2 1
1200.2.h.n 4 4.b odd 2 1
1200.2.h.n 4 5.b even 2 1
1200.2.h.n 4 60.h even 2 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}^{2} - 24$$ $$T_{13}^{2} - 24$$ $$T_{23} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + 2 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -24 + T^{2} )^{2}$$
$13$ $$( -24 + T^{2} )^{2}$$
$17$ $$( 12 + T^{2} )^{2}$$
$19$ $$( 12 + T^{2} )^{2}$$
$23$ $$( -6 + T )^{4}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$( 12 + T^{2} )^{2}$$
$37$ $$( -24 + T^{2} )^{2}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$( 72 + T^{2} )^{2}$$
$47$ $$( 6 + T )^{4}$$
$53$ $$( 108 + T^{2} )^{2}$$
$59$ $$( -24 + T^{2} )^{2}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$( 72 + T^{2} )^{2}$$
$71$ $$( -96 + T^{2} )^{2}$$
$73$ $$( -96 + T^{2} )^{2}$$
$79$ $$( 108 + T^{2} )^{2}$$
$83$ $$( -6 + T )^{4}$$
$89$ $$( 32 + T^{2} )^{2}$$
$97$ $$T^{4}$$