# Properties

 Label 1200.3.bg.f 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -2 + 2 \beta_{2} + \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -2 + 2 \beta_{2} + \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( 12 + 5 \beta_{1} + 12 \beta_{2} ) q^{13} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{19} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{21} + ( -18 + 6 \beta_{1} - 18 \beta_{2} ) q^{23} + 3 \beta_{3} q^{27} + ( -14 \beta_{1} - 2 \beta_{2} - 14 \beta_{3} ) q^{29} + ( -11 - 6 \beta_{1} + 6 \beta_{3} ) q^{31} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{33} + ( 28 - 28 \beta_{2} + 12 \beta_{3} ) q^{37} + ( 12 \beta_{1} + 15 \beta_{2} + 12 \beta_{3} ) q^{39} + ( -8 + 2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( -26 + 19 \beta_{1} - 26 \beta_{2} ) q^{43} + ( -20 + 20 \beta_{2} + 14 \beta_{3} ) q^{47} + ( -4 \beta_{1} + 38 \beta_{2} - 4 \beta_{3} ) q^{49} + ( -6 + 2 \beta_{1} - 2 \beta_{3} ) q^{51} + ( 26 + 32 \beta_{1} + 26 \beta_{2} ) q^{53} + ( -6 + 6 \beta_{2} + 5 \beta_{3} ) q^{57} + ( 32 \beta_{1} + 2 \beta_{2} + 32 \beta_{3} ) q^{59} + ( -45 + 24 \beta_{1} - 24 \beta_{3} ) q^{61} + ( -6 - 3 \beta_{1} - 6 \beta_{2} ) q^{63} + ( -66 + 66 \beta_{2} + 15 \beta_{3} ) q^{67} + ( -18 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} ) q^{69} + ( -64 - 14 \beta_{1} + 14 \beta_{3} ) q^{71} + ( 28 + 20 \beta_{1} + 28 \beta_{2} ) q^{73} + ( -10 + 10 \beta_{2} + 10 \beta_{3} ) q^{77} + ( -4 \beta_{1} + 78 \beta_{2} - 4 \beta_{3} ) q^{79} -9 q^{81} + ( 4 + 70 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 42 - 42 \beta_{2} - 2 \beta_{3} ) q^{87} + ( -28 \beta_{1} + 28 \beta_{2} - 28 \beta_{3} ) q^{89} + ( -63 - 22 \beta_{1} + 22 \beta_{3} ) q^{91} + ( -18 - 11 \beta_{1} - 18 \beta_{2} ) q^{93} + ( 80 - 80 \beta_{2} - 21 \beta_{3} ) q^{97} + ( 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7} + O(q^{10})$$ $$4 q - 8 q^{7} + 8 q^{11} + 48 q^{13} + 8 q^{17} - 12 q^{21} - 72 q^{23} - 44 q^{31} + 24 q^{33} + 112 q^{37} - 32 q^{41} - 104 q^{43} - 80 q^{47} - 24 q^{51} + 104 q^{53} - 24 q^{57} - 180 q^{61} - 24 q^{63} - 264 q^{67} - 256 q^{71} + 112 q^{73} - 40 q^{77} - 36 q^{81} + 16 q^{83} + 168 q^{87} - 252 q^{91} - 72 q^{93} + 320 q^{97} + O(q^{100})$$

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 −0.775255 + 0.775255i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −3.22474 + 3.22474i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −0.775255 0.775255i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 −3.22474 3.22474i 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.f 4
4.b odd 2 1 600.3.u.d yes 4
5.b even 2 1 1200.3.bg.l 4
5.c odd 4 1 inner 1200.3.bg.f 4
5.c odd 4 1 1200.3.bg.l 4
12.b even 2 1 1800.3.v.m 4
20.d odd 2 1 600.3.u.c 4
20.e even 4 1 600.3.u.c 4
20.e even 4 1 600.3.u.d yes 4
60.h even 2 1 1800.3.v.l 4
60.l odd 4 1 1800.3.v.l 4
60.l odd 4 1 1800.3.v.m 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 8 T_{7}^{3} + 32 T_{7}^{2} + 40 T_{7} + 25$$ acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$25 + 40 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$11$ $$( -20 - 4 T + T^{2} )^{2}$$
$13$ $$45369 - 10224 T + 1152 T^{2} - 48 T^{3} + T^{4}$$
$17$ $$16 + 32 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$1 + 98 T^{2} + T^{4}$$
$23$ $$291600 + 38880 T + 2592 T^{2} + 72 T^{3} + T^{4}$$
$29$ $$1373584 + 2360 T^{2} + T^{4}$$
$31$ $$( -95 + 22 T + T^{2} )^{2}$$
$37$ $$1290496 - 127232 T + 6272 T^{2} - 112 T^{3} + T^{4}$$
$41$ $$( 40 + 16 T + T^{2} )^{2}$$
$43$ $$72361 + 27976 T + 5408 T^{2} + 104 T^{3} + T^{4}$$
$47$ $$44944 + 16960 T + 3200 T^{2} + 80 T^{3} + T^{4}$$
$53$ $$2958400 + 178880 T + 5408 T^{2} - 104 T^{3} + T^{4}$$
$59$ $$37699600 + 12296 T^{2} + T^{4}$$
$61$ $$( -1431 + 90 T + T^{2} )^{2}$$
$67$ $$64593369 + 2121768 T + 34848 T^{2} + 264 T^{3} + T^{4}$$
$71$ $$( 2920 + 128 T + T^{2} )^{2}$$
$73$ $$135424 - 41216 T + 6272 T^{2} - 112 T^{3} + T^{4}$$
$79$ $$35856144 + 12360 T^{2} + T^{4}$$
$83$ $$215150224 + 234688 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$89$ $$15366400 + 10976 T^{2} + T^{4}$$
$97$ $$131721529 - 3672640 T + 51200 T^{2} - 320 T^{3} + T^{4}$$