# Properties

 Label 1200.2.h.n 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: $$x^{4} - 2x^{2} + 4$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + ( - 2 \beta_1 - 1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + (-2*b1 - 1) * q^9 $$q + ( - \beta_1 + 1) q^{3} + ( - 2 \beta_1 - 1) q^{9} - \beta_{3} q^{11} - \beta_{3} q^{13} + \beta_{2} q^{17} - \beta_{2} q^{19} - 6 q^{23} + ( - \beta_1 - 5) q^{27} + 2 \beta_1 q^{29} - \beta_{2} q^{31} + ( - \beta_{3} + 2 \beta_{2}) q^{33} - \beta_{3} q^{37} + ( - \beta_{3} + 2 \beta_{2}) q^{39} - 4 \beta_1 q^{41} - 6 \beta_1 q^{43} + 6 q^{47} + 7 q^{49} + (\beta_{3} + \beta_{2}) q^{51} + 3 \beta_{2} q^{53} + ( - \beta_{3} - \beta_{2}) q^{57} + \beta_{3} q^{59} - 2 q^{61} - 6 \beta_1 q^{67} + (6 \beta_1 - 6) q^{69} + 2 \beta_{3} q^{71} - 2 \beta_{3} q^{73} - 3 \beta_{2} q^{79} + (4 \beta_1 - 7) q^{81} - 6 q^{83} + (2 \beta_1 + 4) q^{87} - 4 \beta_1 q^{89} + ( - \beta_{3} - \beta_{2}) q^{93} + (\beta_{3} + 4 \beta_{2}) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + (-2*b1 - 1) * q^9 - b3 * q^11 - b3 * q^13 + b2 * q^17 - b2 * q^19 - 6 * q^23 + (-b1 - 5) * q^27 + 2*b1 * q^29 - b2 * q^31 + (-b3 + 2*b2) * q^33 - b3 * q^37 + (-b3 + 2*b2) * q^39 - 4*b1 * q^41 - 6*b1 * q^43 + 6 * q^47 + 7 * q^49 + (b3 + b2) * q^51 + 3*b2 * q^53 + (-b3 - b2) * q^57 + b3 * q^59 - 2 * q^61 - 6*b1 * q^67 + (6*b1 - 6) * q^69 + 2*b3 * q^71 - 2*b3 * q^73 - 3*b2 * q^79 + (4*b1 - 7) * q^81 - 6 * q^83 + (2*b1 + 4) * q^87 - 4*b1 * q^89 + (-b3 - b2) * q^93 + (b3 + 4*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^9 $$4 q + 4 q^{3} - 4 q^{9} - 24 q^{23} - 20 q^{27} + 24 q^{47} + 28 q^{49} - 8 q^{61} - 24 q^{69} - 28 q^{81} - 24 q^{83} + 16 q^{87}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 2$$ 2*v^2 - 2 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 4\nu$$ -v^3 + 4*v
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_1 ) / 4$$ (b3 + 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 2 ) / 2$$ (b2 + 2) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_1$$ 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
 1.22474 + 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.n 4
3.b odd 2 1 1200.2.h.j 4
4.b odd 2 1 1200.2.h.j 4
5.b even 2 1 1200.2.h.j 4
5.c odd 4 2 240.2.o.b 8
12.b even 2 1 inner 1200.2.h.n 4
15.d odd 2 1 inner 1200.2.h.n 4
15.e even 4 2 240.2.o.b 8
20.d odd 2 1 inner 1200.2.h.n 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.j 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 3.b odd 2 1
1200.2.h.j 4 4.b odd 2 1
1200.2.h.j 4 5.b even 2 1
1200.2.h.j 4 60.h even 2 1
1200.2.h.n 4 1.a even 1 1 trivial
1200.2.h.n 4 12.b even 2 1 inner
1200.2.h.n 4 15.d odd 2 1 inner
1200.2.h.n 4 20.d odd 2 1 inner

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11}^{2} - 24$$ T11^2 - 24 $$T_{13}^{2} - 24$$ T13^2 - 24 $$T_{23} + 6$$ T23 + 6

## Hecke characteristic polynomials

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