# Properties

 Label 1200.3.bg.a Level $1200$ Weight $3$ Character orbit 1200.bg Analytic conductor $32.698$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.bg (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -6 + 6 \beta_{2} - \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -6 + 6 \beta_{2} - \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( -6 - 6 \beta_{1} + 6 \beta_{3} ) q^{11} + ( -12 + 3 \beta_{1} - 12 \beta_{2} ) q^{13} + ( 6 - 6 \beta_{2} + 6 \beta_{3} ) q^{17} + ( -6 \beta_{1} - 19 \beta_{2} - 6 \beta_{3} ) q^{19} + ( -3 + 6 \beta_{1} - 6 \beta_{3} ) q^{21} + ( -6 + 18 \beta_{1} - 6 \beta_{2} ) q^{23} -3 \beta_{3} q^{27} + ( -6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{29} + ( 5 + 18 \beta_{1} - 18 \beta_{3} ) q^{31} + ( 18 + 6 \beta_{1} + 18 \beta_{2} ) q^{33} + ( -12 + 12 \beta_{2} + 20 \beta_{3} ) q^{37} + ( 12 \beta_{1} - 9 \beta_{2} + 12 \beta_{3} ) q^{39} + ( 48 - 6 \beta_{1} + 6 \beta_{3} ) q^{41} + ( 18 + 5 \beta_{1} + 18 \beta_{2} ) q^{43} + ( 36 - 36 \beta_{2} + 18 \beta_{3} ) q^{47} + ( 12 \beta_{1} - 26 \beta_{2} + 12 \beta_{3} ) q^{49} + ( 18 - 6 \beta_{1} + 6 \beta_{3} ) q^{51} + ( 30 + 24 \beta_{1} + 30 \beta_{2} ) q^{53} + ( -18 + 18 \beta_{2} + 19 \beta_{3} ) q^{57} -30 \beta_{2} q^{59} + ( 11 - 24 \beta_{1} + 24 \beta_{3} ) q^{61} + ( -18 + 3 \beta_{1} - 18 \beta_{2} ) q^{63} + ( -6 + 6 \beta_{2} + 9 \beta_{3} ) q^{67} + ( 6 \beta_{1} - 54 \beta_{2} + 6 \beta_{3} ) q^{69} + ( 24 - 6 \beta_{1} + 6 \beta_{3} ) q^{71} + ( -12 + 28 \beta_{1} - 12 \beta_{2} ) q^{73} + ( 18 - 18 \beta_{2} - 66 \beta_{3} ) q^{77} + ( 12 \beta_{1} - 2 \beta_{2} + 12 \beta_{3} ) q^{79} -9 q^{81} + ( 12 + 42 \beta_{1} + 12 \beta_{2} ) q^{83} + ( -18 + 18 \beta_{2} - 6 \beta_{3} ) q^{87} + ( -12 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{89} + ( 153 - 30 \beta_{1} + 30 \beta_{3} ) q^{91} + ( -54 - 5 \beta_{1} - 54 \beta_{2} ) q^{93} + ( 48 - 48 \beta_{2} + 5 \beta_{3} ) q^{97} + ( -18 \beta_{1} - 18 \beta_{2} - 18 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 24 q^{7} + O(q^{10})$$ $$4 q - 24 q^{7} - 24 q^{11} - 48 q^{13} + 24 q^{17} - 12 q^{21} - 24 q^{23} + 20 q^{31} + 72 q^{33} - 48 q^{37} + 192 q^{41} + 72 q^{43} + 144 q^{47} + 72 q^{51} + 120 q^{53} - 72 q^{57} + 44 q^{61} - 72 q^{63} - 24 q^{67} + 96 q^{71} - 48 q^{73} + 72 q^{77} - 36 q^{81} + 48 q^{83} - 72 q^{87} + 612 q^{91} - 216 q^{93} + 192 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
0 −1.22474 1.22474i 0 0 0 −4.77526 + 4.77526i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −7.22474 + 7.22474i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −4.77526 4.77526i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 −7.22474 7.22474i 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.bg.a 4
4.b odd 2 1 150.3.f.c yes 4
5.b even 2 1 1200.3.bg.p 4
5.c odd 4 1 inner 1200.3.bg.a 4
5.c odd 4 1 1200.3.bg.p 4
12.b even 2 1 450.3.g.g 4
20.d odd 2 1 150.3.f.a 4
20.e even 4 1 150.3.f.a 4
20.e even 4 1 150.3.f.c yes 4
60.h even 2 1 450.3.g.h 4
60.l odd 4 1 450.3.g.g 4
60.l odd 4 1 450.3.g.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.f.a 4 20.d odd 2 1
150.3.f.a 4 20.e even 4 1
150.3.f.c yes 4 4.b odd 2 1
150.3.f.c yes 4 20.e even 4 1
450.3.g.g 4 12.b even 2 1
450.3.g.g 4 60.l odd 4 1
450.3.g.h 4 60.h even 2 1
450.3.g.h 4 60.l odd 4 1
1200.3.bg.a 4 1.a even 1 1 trivial
1200.3.bg.a 4 5.c odd 4 1 inner
1200.3.bg.p 4 5.b even 2 1
1200.3.bg.p 4 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 24 T_{7}^{3} + 288 T_{7}^{2} + 1656 T_{7} + 4761$$ acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$4761 + 1656 T + 288 T^{2} + 24 T^{3} + T^{4}$$
$11$ $$( -180 + 12 T + T^{2} )^{2}$$
$13$ $$68121 + 12528 T + 1152 T^{2} + 48 T^{3} + T^{4}$$
$17$ $$1296 + 864 T + 288 T^{2} - 24 T^{3} + T^{4}$$
$19$ $$21025 + 1154 T^{2} + T^{4}$$
$23$ $$810000 - 21600 T + 288 T^{2} + 24 T^{3} + T^{4}$$
$29$ $$32400 + 504 T^{2} + T^{4}$$
$31$ $$( -1919 - 10 T + T^{2} )^{2}$$
$37$ $$831744 - 43776 T + 1152 T^{2} + 48 T^{3} + T^{4}$$
$41$ $$( 2088 - 96 T + T^{2} )^{2}$$
$43$ $$328329 - 41256 T + 2592 T^{2} - 72 T^{3} + T^{4}$$
$47$ $$2624400 - 233280 T + 10368 T^{2} - 144 T^{3} + T^{4}$$
$53$ $$5184 - 8640 T + 7200 T^{2} - 120 T^{3} + T^{4}$$
$59$ $$( 900 + T^{2} )^{2}$$
$61$ $$( -3335 - 22 T + T^{2} )^{2}$$
$67$ $$29241 - 4104 T + 288 T^{2} + 24 T^{3} + T^{4}$$
$71$ $$( 360 - 48 T + T^{2} )^{2}$$
$73$ $$4260096 - 99072 T + 1152 T^{2} + 48 T^{3} + T^{4}$$
$79$ $$739600 + 1736 T^{2} + T^{4}$$
$83$ $$25040016 + 240192 T + 1152 T^{2} - 48 T^{3} + T^{4}$$
$89$ $$518400 + 2016 T^{2} + T^{4}$$
$97$ $$20548089 - 870336 T + 18432 T^{2} - 192 T^{3} + T^{4}$$