# Properties

 Label 1200.3.bg.c 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 600) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -4 + 4 \beta_{2} + 3 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -4 + 4 \beta_{2} + 3 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( 2 - 4 \beta_{1} + 4 \beta_{3} ) q^{11} -3 \beta_{1} q^{13} + ( 4 - 4 \beta_{2} + 6 \beta_{3} ) q^{17} + ( 12 \beta_{1} - \beta_{2} + 12 \beta_{3} ) q^{19} + ( 9 + 4 \beta_{1} - 4 \beta_{3} ) q^{21} + ( -12 - 2 \beta_{1} - 12 \beta_{2} ) q^{23} -3 \beta_{3} q^{27} + ( -12 \beta_{1} - 14 \beta_{2} - 12 \beta_{3} ) q^{29} + ( 31 + 4 \beta_{1} - 4 \beta_{3} ) q^{31} + ( 12 - 2 \beta_{1} + 12 \beta_{2} ) q^{33} + ( 8 - 8 \beta_{2} - 4 \beta_{3} ) q^{37} + 9 \beta_{2} q^{39} + ( 28 + 4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -28 - 7 \beta_{1} - 28 \beta_{2} ) q^{43} + ( -40 + 40 \beta_{2} + 10 \beta_{3} ) q^{47} + ( -24 \beta_{1} - 10 \beta_{2} - 24 \beta_{3} ) q^{49} + ( 18 - 4 \beta_{1} + 4 \beta_{3} ) q^{51} + ( 52 - 4 \beta_{1} + 52 \beta_{2} ) q^{53} + ( 36 - 36 \beta_{2} + \beta_{3} ) q^{57} -82 \beta_{2} q^{59} + ( 75 - 8 \beta_{1} + 8 \beta_{3} ) q^{61} + ( -12 - 9 \beta_{1} - 12 \beta_{2} ) q^{63} + ( 36 - 36 \beta_{2} - 19 \beta_{3} ) q^{67} + ( 12 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} ) q^{69} + ( 68 - 4 \beta_{1} + 4 \beta_{3} ) q^{71} + ( 56 - 28 \beta_{1} + 56 \beta_{2} ) q^{73} + ( 28 - 28 \beta_{2} - 26 \beta_{3} ) q^{77} + ( 40 \beta_{1} - 6 \beta_{2} + 40 \beta_{3} ) q^{79} -9 q^{81} + ( -40 - 14 \beta_{1} - 40 \beta_{2} ) q^{83} + ( -36 + 36 \beta_{2} + 14 \beta_{3} ) q^{87} + ( 8 \beta_{1} - 140 \beta_{2} + 8 \beta_{3} ) q^{89} + ( 27 + 12 \beta_{1} - 12 \beta_{3} ) q^{91} + ( -12 - 31 \beta_{1} - 12 \beta_{2} ) q^{93} + ( -80 + 80 \beta_{2} + 19 \beta_{3} ) q^{97} + ( -12 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{7} + O(q^{10})$$ $$4 q - 16 q^{7} + 8 q^{11} + 16 q^{17} + 36 q^{21} - 48 q^{23} + 124 q^{31} + 48 q^{33} + 32 q^{37} + 112 q^{41} - 112 q^{43} - 160 q^{47} + 72 q^{51} + 208 q^{53} + 144 q^{57} + 300 q^{61} - 48 q^{63} + 144 q^{67} + 272 q^{71} + 224 q^{73} + 112 q^{77} - 36 q^{81} - 160 q^{83} - 144 q^{87} + 108 q^{91} - 48 q^{93} - 320 q^{97} + O(q^{100})$$

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 −7.67423 + 7.67423i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −0.325765 + 0.325765i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −7.67423 7.67423i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 −0.325765 0.325765i 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.c 4
4.b odd 2 1 600.3.u.g yes 4
5.b even 2 1 1200.3.bg.n 4
5.c odd 4 1 inner 1200.3.bg.c 4
5.c odd 4 1 1200.3.bg.n 4
12.b even 2 1 1800.3.v.p 4
20.d odd 2 1 600.3.u.b 4
20.e even 4 1 600.3.u.b 4
20.e even 4 1 600.3.u.g yes 4
60.h even 2 1 1800.3.v.i 4
60.l odd 4 1 1800.3.v.i 4
60.l odd 4 1 1800.3.v.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.u.b 4 20.d odd 2 1
600.3.u.b 4 20.e even 4 1
600.3.u.g yes 4 4.b odd 2 1
600.3.u.g yes 4 20.e even 4 1
1200.3.bg.c 4 1.a even 1 1 trivial
1200.3.bg.c 4 5.c odd 4 1 inner
1200.3.bg.n 4 5.b even 2 1
1200.3.bg.n 4 5.c odd 4 1
1800.3.v.i 4 60.h even 2 1
1800.3.v.i 4 60.l odd 4 1
1800.3.v.p 4 12.b even 2 1
1800.3.v.p 4 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 16 T_{7}^{3} + 128 T_{7}^{2} + 80 T_{7} + 25$$ acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$25 + 80 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$11$ $$( -92 - 4 T + T^{2} )^{2}$$
$13$ $$729 + T^{4}$$
$17$ $$5776 + 1216 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$19$ $$744769 + 1730 T^{2} + T^{4}$$
$23$ $$76176 + 13248 T + 1152 T^{2} + 48 T^{3} + T^{4}$$
$29$ $$446224 + 2120 T^{2} + T^{4}$$
$31$ $$( 865 - 62 T + T^{2} )^{2}$$
$37$ $$6400 - 2560 T + 512 T^{2} - 32 T^{3} + T^{4}$$
$41$ $$( 688 - 56 T + T^{2} )^{2}$$
$43$ $$2019241 + 159152 T + 6272 T^{2} + 112 T^{3} + T^{4}$$
$47$ $$8410000 + 464000 T + 12800 T^{2} + 160 T^{3} + T^{4}$$
$53$ $$28729600 - 1114880 T + 21632 T^{2} - 208 T^{3} + T^{4}$$
$59$ $$( 6724 + T^{2} )^{2}$$
$61$ $$( 5241 - 150 T + T^{2} )^{2}$$
$67$ $$2277081 - 217296 T + 10368 T^{2} - 144 T^{3} + T^{4}$$
$71$ $$( 4528 - 136 T + T^{2} )^{2}$$
$73$ $$15366400 - 878080 T + 25088 T^{2} - 224 T^{3} + T^{4}$$
$79$ $$91470096 + 19272 T^{2} + T^{4}$$
$83$ $$6822544 + 417920 T + 12800 T^{2} + 160 T^{3} + T^{4}$$
$89$ $$369254656 + 39968 T^{2} + T^{4}$$
$97$ $$137288089 + 3749440 T + 51200 T^{2} + 320 T^{3} + T^{4}$$