# Properties

 Label 1200.2.h.l Level $1200$ Weight $2$ Character orbit 1200.h Analytic conductor $9.582$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# 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: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q - b1 * q^3 + (b3 + b1) * q^7 + (b2 + 2) * q^9 $$q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + (\beta_{2} + 2) q^{9} + ( - 3 \beta_{2} + 3) q^{21} + (\beta_{3} - 5 \beta_1) q^{23} + ( - \beta_{3} - 2 \beta_1) q^{27} - 4 \beta_{2} q^{29} + 2 \beta_{2} q^{41} + (3 \beta_{3} + 3 \beta_1) q^{43} + (\beta_{3} - 5 \beta_1) q^{47} - 11 q^{49} - 8 q^{61} + (3 \beta_{3} - 3 \beta_1) q^{63} + ( - \beta_{3} - \beta_1) q^{67} + (3 \beta_{2} + 15) q^{69} + (4 \beta_{2} - 1) q^{81} + (\beta_{3} - 5 \beta_1) q^{83} + 4 \beta_{3} q^{87} + 8 \beta_{2} q^{89}+O(q^{100})$$ q - b1 * q^3 + (b3 + b1) * q^7 + (b2 + 2) * q^9 + (-3*b2 + 3) * q^21 + (b3 - 5*b1) * q^23 + (-b3 - 2*b1) * q^27 - 4*b2 * q^29 + 2*b2 * q^41 + (3*b3 + 3*b1) * q^43 + (b3 - 5*b1) * q^47 - 11 * q^49 - 8 * q^61 + (3*b3 - 3*b1) * q^63 + (-b3 - b1) * q^67 + (3*b2 + 15) * q^69 + (4*b2 - 1) * q^81 + (b3 - 5*b1) * q^83 + 4*b3 * q^87 + 8*b2 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{9}+O(q^{10})$$ 4 * q + 8 * q^9 $$4 q + 8 q^{9} + 12 q^{21} - 44 q^{49} - 32 q^{61} + 60 q^{69} - 4 q^{81}+O(q^{100})$$ 4 * q + 8 * q^9 + 12 * q^21 - 44 * q^49 - 32 * q^61 + 60 * q^69 - 4 * q^81

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu$$ v^3 - 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_1$$ b3 + 2*b1

## 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
 1.58114 + 0.707107i 1.58114 − 0.707107i −1.58114 + 0.707107i −1.58114 − 0.707107i
0 −1.58114 0.707107i 0 0 0 4.24264i 0 2.00000 + 2.23607i 0
1151.2 0 −1.58114 + 0.707107i 0 0 0 4.24264i 0 2.00000 2.23607i 0
1151.3 0 1.58114 0.707107i 0 0 0 4.24264i 0 2.00000 2.23607i 0
1151.4 0 1.58114 + 0.707107i 0 0 0 4.24264i 0 2.00000 + 2.23607i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.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.h.l 4
3.b odd 2 1 inner 1200.2.h.l 4
4.b odd 2 1 inner 1200.2.h.l 4
5.b even 2 1 inner 1200.2.h.l 4
5.c odd 4 2 240.2.o.a 4
12.b even 2 1 inner 1200.2.h.l 4
15.d odd 2 1 inner 1200.2.h.l 4
15.e even 4 2 240.2.o.a 4
20.d odd 2 1 CM 1200.2.h.l 4
20.e even 4 2 240.2.o.a 4
40.i odd 4 2 960.2.o.b 4
40.k even 4 2 960.2.o.b 4
60.h even 2 1 inner 1200.2.h.l 4
60.l odd 4 2 240.2.o.a 4
120.q odd 4 2 960.2.o.b 4
120.w even 4 2 960.2.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.o.a 4 5.c odd 4 2
240.2.o.a 4 15.e even 4 2
240.2.o.a 4 20.e even 4 2
240.2.o.a 4 60.l odd 4 2
960.2.o.b 4 40.i odd 4 2
960.2.o.b 4 40.k even 4 2
960.2.o.b 4 120.q odd 4 2
960.2.o.b 4 120.w even 4 2
1200.2.h.l 4 1.a even 1 1 trivial
1200.2.h.l 4 3.b odd 2 1 inner
1200.2.h.l 4 4.b odd 2 1 inner
1200.2.h.l 4 5.b even 2 1 inner
1200.2.h.l 4 12.b even 2 1 inner
1200.2.h.l 4 15.d odd 2 1 inner
1200.2.h.l 4 20.d odd 2 1 CM
1200.2.h.l 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 $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 18$$ T7^2 + 18 $$T_{11}$$ T11 $$T_{13}$$ T13 $$T_{23}^{2} - 90$$ T23^2 - 90

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 18)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} - 90)^{2}$$
$29$ $$(T^{2} + 80)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 20)^{2}$$
$43$ $$(T^{2} + 162)^{2}$$
$47$ $$(T^{2} - 90)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T + 8)^{4}$$
$67$ $$(T^{2} + 18)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 90)^{2}$$
$89$ $$(T^{2} + 320)^{2}$$
$97$ $$T^{4}$$