# Properties

 Label 1200.3.bg.d 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{7} -3 \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{7} -3 \beta_{2} q^{9} + ( 4 + 4 \beta_{1} - 4 \beta_{3} ) q^{11} + ( -3 + 3 \beta_{2} + 8 \beta_{3} ) q^{13} + ( -11 - 4 \beta_{1} - 11 \beta_{2} ) q^{17} + ( -4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 12 - 4 \beta_{1} + 4 \beta_{3} ) q^{21} + ( -4 + 4 \beta_{2} - 12 \beta_{3} ) q^{23} -3 \beta_{1} q^{27} + ( -4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} ) q^{29} + ( 20 + 8 \beta_{1} - 8 \beta_{3} ) q^{31} + ( 12 - 12 \beta_{2} - 4 \beta_{3} ) q^{33} + ( 27 + 27 \beta_{2} ) q^{37} + ( 3 \beta_{1} + 24 \beta_{2} + 3 \beta_{3} ) q^{39} + ( 8 + 4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( 12 - 12 \beta_{2} + 20 \beta_{3} ) q^{43} + ( 24 - 12 \beta_{1} + 24 \beta_{2} ) q^{47} + ( -32 \beta_{1} + 31 \beta_{2} - 32 \beta_{3} ) q^{49} + ( -12 - 11 \beta_{1} + 11 \beta_{3} ) q^{51} + ( -25 + 25 \beta_{2} - 36 \beta_{3} ) q^{53} + ( -12 - 16 \beta_{1} - 12 \beta_{2} ) q^{57} -20 \beta_{2} q^{59} + ( -24 + 16 \beta_{1} - 16 \beta_{3} ) q^{61} + ( -12 + 12 \beta_{2} - 12 \beta_{3} ) q^{63} + ( -4 - 36 \beta_{1} - 4 \beta_{2} ) q^{67} + ( 4 \beta_{1} - 36 \beta_{2} + 4 \beta_{3} ) q^{69} + ( -16 + 4 \beta_{1} - 4 \beta_{3} ) q^{71} + ( 47 - 47 \beta_{2} + 8 \beta_{3} ) q^{73} + ( 32 - 16 \beta_{1} + 32 \beta_{2} ) q^{77} + ( -52 \beta_{1} + 12 \beta_{2} - 52 \beta_{3} ) q^{79} -9 q^{81} + ( 48 - 48 \beta_{2} - 28 \beta_{3} ) q^{83} + ( -12 - 16 \beta_{1} - 12 \beta_{2} ) q^{87} + ( 12 \beta_{1} + 88 \beta_{2} + 12 \beta_{3} ) q^{89} + ( -72 + 20 \beta_{1} - 20 \beta_{3} ) q^{91} + ( 24 - 24 \beta_{2} - 20 \beta_{3} ) q^{93} + ( -33 - 40 \beta_{1} - 33 \beta_{2} ) q^{97} + ( -12 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{7} + O(q^{10})$$ $$4q - 16q^{7} + 16q^{11} - 12q^{13} - 44q^{17} + 48q^{21} - 16q^{23} + 80q^{31} + 48q^{33} + 108q^{37} + 32q^{41} + 48q^{43} + 96q^{47} - 48q^{51} - 100q^{53} - 48q^{57} - 96q^{61} - 48q^{63} - 16q^{67} - 64q^{71} + 188q^{73} + 128q^{77} - 36q^{81} + 192q^{83} - 48q^{87} - 288q^{91} + 96q^{93} - 132q^{97} + O(q^{100})$$

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 −8.89898 + 8.89898i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 0.898979 0.898979i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −8.89898 8.89898i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 0.898979 + 0.898979i 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.d 4
4.b odd 2 1 150.3.f.b 4
5.b even 2 1 240.3.bg.b 4
5.c odd 4 1 240.3.bg.b 4
5.c odd 4 1 inner 1200.3.bg.d 4
12.b even 2 1 450.3.g.j 4
15.d odd 2 1 720.3.bh.i 4
15.e even 4 1 720.3.bh.i 4
20.d odd 2 1 30.3.f.a 4
20.e even 4 1 30.3.f.a 4
20.e even 4 1 150.3.f.b 4
40.e odd 2 1 960.3.bg.e 4
40.f even 2 1 960.3.bg.g 4
40.i odd 4 1 960.3.bg.g 4
40.k even 4 1 960.3.bg.e 4
60.h even 2 1 90.3.g.d 4
60.l odd 4 1 90.3.g.d 4
60.l odd 4 1 450.3.g.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.f.a 4 20.d odd 2 1
30.3.f.a 4 20.e even 4 1
90.3.g.d 4 60.h even 2 1
90.3.g.d 4 60.l odd 4 1
150.3.f.b 4 4.b odd 2 1
150.3.f.b 4 20.e even 4 1
240.3.bg.b 4 5.b even 2 1
240.3.bg.b 4 5.c odd 4 1
450.3.g.j 4 12.b even 2 1
450.3.g.j 4 60.l odd 4 1
720.3.bh.i 4 15.d odd 2 1
720.3.bh.i 4 15.e even 4 1
960.3.bg.e 4 40.e odd 2 1
960.3.bg.e 4 40.k even 4 1
960.3.bg.g 4 40.f even 2 1
960.3.bg.g 4 40.i odd 4 1
1200.3.bg.d 4 1.a even 1 1 trivial
1200.3.bg.d 4 5.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 16 T_{7}^{3} + 128 T_{7}^{2} - 256 T_{7} + 256$$ acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$256 - 256 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$11$ $$( -80 - 8 T + T^{2} )^{2}$$
$13$ $$30276 - 2088 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$17$ $$37636 + 8536 T + 968 T^{2} + 44 T^{3} + T^{4}$$
$19$ $$25600 + 704 T^{2} + T^{4}$$
$23$ $$160000 - 6400 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$29$ $$25600 + 704 T^{2} + T^{4}$$
$31$ $$( 16 - 40 T + T^{2} )^{2}$$
$37$ $$( 1458 - 54 T + T^{2} )^{2}$$
$41$ $$( -32 - 16 T + T^{2} )^{2}$$
$43$ $$831744 + 43776 T + 1152 T^{2} - 48 T^{3} + T^{4}$$
$47$ $$518400 - 69120 T + 4608 T^{2} - 96 T^{3} + T^{4}$$
$53$ $$6959044 - 263800 T + 5000 T^{2} + 100 T^{3} + T^{4}$$
$59$ $$( 400 + T^{2} )^{2}$$
$61$ $$( -960 + 48 T + T^{2} )^{2}$$
$67$ $$14868736 - 61696 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$71$ $$( 160 + 32 T + T^{2} )^{2}$$
$73$ $$17859076 - 794488 T + 17672 T^{2} - 188 T^{3} + T^{4}$$
$79$ $$258566400 + 32736 T^{2} + T^{4}$$
$83$ $$5089536 - 433152 T + 18432 T^{2} - 192 T^{3} + T^{4}$$
$89$ $$47334400 + 17216 T^{2} + T^{4}$$
$97$ $$6874884 - 346104 T + 8712 T^{2} + 132 T^{3} + T^{4}$$