# Properties

 Label 1200.2.h.k 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 + (b3 + b1) * q^7 + b2 * q^9 $$q + \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9} + (2 \beta_{3} - 2 \beta_1) q^{11} + 2 q^{13} + 2 \beta_{2} q^{17} + ( - 2 \beta_{3} - 2 \beta_1) q^{19} + (\beta_{2} - 3) q^{21} + (\beta_{3} - \beta_1) q^{23} + 3 \beta_{3} q^{27} + (4 \beta_{3} + 4 \beta_1) q^{31} + ( - 2 \beta_{2} - 6) q^{33} - 2 q^{37} + 2 \beta_1 q^{39} - 2 \beta_{2} q^{41} + ( - \beta_{3} - \beta_1) q^{43} + (5 \beta_{3} - 5 \beta_1) q^{47} + q^{49} + 6 \beta_{3} q^{51} + 2 \beta_{2} q^{53} + ( - 2 \beta_{2} + 6) q^{57} + ( - 4 \beta_{3} + 4 \beta_1) q^{59} + 8 q^{61} + (3 \beta_{3} - 3 \beta_1) q^{63} + (3 \beta_{3} + 3 \beta_1) q^{67} + ( - \beta_{2} - 3) q^{69} + (2 \beta_{3} - 2 \beta_1) q^{71} + 14 q^{73} - 4 \beta_{2} q^{77} + (2 \beta_{3} + 2 \beta_1) q^{79} - 9 q^{81} + ( - 3 \beta_{3} + 3 \beta_1) q^{83} - 4 \beta_{2} q^{89} + (2 \beta_{3} + 2 \beta_1) q^{91} + (4 \beta_{2} - 12) q^{93} - 10 q^{97} + ( - 6 \beta_{3} - 6 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b3 + b1) * q^7 + b2 * q^9 + (2*b3 - 2*b1) * q^11 + 2 * q^13 + 2*b2 * q^17 + (-2*b3 - 2*b1) * q^19 + (b2 - 3) * q^21 + (b3 - b1) * q^23 + 3*b3 * q^27 + (4*b3 + 4*b1) * q^31 + (-2*b2 - 6) * q^33 - 2 * q^37 + 2*b1 * q^39 - 2*b2 * q^41 + (-b3 - b1) * q^43 + (5*b3 - 5*b1) * q^47 + q^49 + 6*b3 * q^51 + 2*b2 * q^53 + (-2*b2 + 6) * q^57 + (-4*b3 + 4*b1) * q^59 + 8 * q^61 + (3*b3 - 3*b1) * q^63 + (3*b3 + 3*b1) * q^67 + (-b2 - 3) * q^69 + (2*b3 - 2*b1) * q^71 + 14 * q^73 - 4*b2 * q^77 + (2*b3 + 2*b1) * q^79 - 9 * q^81 + (-3*b3 + 3*b1) * q^83 - 4*b2 * q^89 + (2*b3 + 2*b1) * q^91 + (4*b2 - 12) * q^93 - 10 * q^97 + (-6*b3 - 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{13} - 12 q^{21} - 24 q^{33} - 8 q^{37} + 4 q^{49} + 24 q^{57} + 32 q^{61} - 12 q^{69} + 56 q^{73} - 36 q^{81} - 48 q^{93} - 40 q^{97}+O(q^{100})$$ 4 * q + 8 * q^13 - 12 * q^21 - 24 * q^33 - 8 * q^37 + 4 * q^49 + 24 * q^57 + 32 * q^61 - 12 * q^69 + 56 * q^73 - 36 * q^81 - 48 * q^93 - 40 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## 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 − 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i 1.22474 + 1.22474i
0 −1.22474 1.22474i 0 0 0 2.44949i 0 3.00000i 0
1151.2 0 −1.22474 + 1.22474i 0 0 0 2.44949i 0 3.00000i 0
1151.3 0 1.22474 1.22474i 0 0 0 2.44949i 0 3.00000i 0
1151.4 0 1.22474 + 1.22474i 0 0 0 2.44949i 0 3.00000i 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.k 4
3.b odd 2 1 inner 1200.2.h.k 4
4.b odd 2 1 inner 1200.2.h.k 4
5.b even 2 1 240.2.h.a 4
5.c odd 4 1 1200.2.o.e 4
5.c odd 4 1 1200.2.o.f 4
12.b even 2 1 inner 1200.2.h.k 4
15.d odd 2 1 240.2.h.a 4
15.e even 4 1 1200.2.o.e 4
15.e even 4 1 1200.2.o.f 4
20.d odd 2 1 240.2.h.a 4
20.e even 4 1 1200.2.o.e 4
20.e even 4 1 1200.2.o.f 4
40.e odd 2 1 960.2.h.c 4
40.f even 2 1 960.2.h.c 4
60.h even 2 1 240.2.h.a 4
60.l odd 4 1 1200.2.o.e 4
60.l odd 4 1 1200.2.o.f 4
120.i odd 2 1 960.2.h.c 4
120.m even 2 1 960.2.h.c 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 6)^{2}$$
$11$ $$(T^{2} - 24)^{2}$$
$13$ $$(T - 2)^{4}$$
$17$ $$(T^{2} + 36)^{2}$$
$19$ $$(T^{2} + 24)^{2}$$
$23$ $$(T^{2} - 6)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 96)^{2}$$
$37$ $$(T + 2)^{4}$$
$41$ $$(T^{2} + 36)^{2}$$
$43$ $$(T^{2} + 6)^{2}$$
$47$ $$(T^{2} - 150)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 96)^{2}$$
$61$ $$(T - 8)^{4}$$
$67$ $$(T^{2} + 54)^{2}$$
$71$ $$(T^{2} - 24)^{2}$$
$73$ $$(T - 14)^{4}$$
$79$ $$(T^{2} + 24)^{2}$$
$83$ $$(T^{2} - 54)^{2}$$
$89$ $$(T^{2} + 144)^{2}$$
$97$ $$(T + 10)^{4}$$