# Properties

 Label 1200.3.bg.h Level $1200$ Weight $3$ Character orbit 1200.bg Analytic conductor $32.698$ 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: $$k$$ $$=$$ $$3$$ 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 300) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + 2 \beta_{1} q^{7} -3 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + 2 \beta_{1} q^{7} -3 \beta_{2} q^{9} -6 q^{11} -10 \beta_{3} q^{13} -12 \beta_{1} q^{17} + 10 \beta_{2} q^{19} -6 q^{21} + 24 \beta_{3} q^{23} + 3 \beta_{1} q^{27} + 48 \beta_{2} q^{29} + 26 q^{31} -6 \beta_{3} q^{33} -26 \beta_{1} q^{37} + 30 \beta_{2} q^{39} + 30 q^{41} + 24 \beta_{3} q^{43} -12 \beta_{1} q^{47} -37 \beta_{2} q^{49} + 36 q^{51} + 12 \beta_{3} q^{53} -10 \beta_{1} q^{57} -78 \beta_{2} q^{59} + 2 q^{61} -6 \beta_{3} q^{63} -52 \beta_{1} q^{67} -72 \beta_{2} q^{69} -120 q^{71} + 68 \beta_{3} q^{73} -12 \beta_{1} q^{77} + 74 \beta_{2} q^{79} -9 q^{81} -36 \beta_{3} q^{83} -48 \beta_{1} q^{87} + 150 \beta_{2} q^{89} + 60 q^{91} + 26 \beta_{3} q^{93} -4 \beta_{1} q^{97} + 18 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 24q^{11} - 24q^{21} + 104q^{31} + 120q^{41} + 144q^{51} + 8q^{61} - 480q^{71} - 36q^{81} + 240q^{91} + O(q^{100})$$

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 2.44949 2.44949i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −2.44949 + 2.44949i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 2.44949 + 2.44949i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 −2.44949 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
5.b even 2 1 inner
5.c odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.bg.h 4
4.b odd 2 1 300.3.k.b 4
5.b even 2 1 inner 1200.3.bg.h 4
5.c odd 4 2 inner 1200.3.bg.h 4
12.b even 2 1 900.3.l.c 4
20.d odd 2 1 300.3.k.b 4
20.e even 4 2 300.3.k.b 4
60.h even 2 1 900.3.l.c 4
60.l odd 4 2 900.3.l.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.k.b 4 4.b odd 2 1
300.3.k.b 4 20.d odd 2 1
300.3.k.b 4 20.e even 4 2
900.3.l.c 4 12.b even 2 1
900.3.l.c 4 60.h even 2 1
900.3.l.c 4 60.l odd 4 2
1200.3.bg.h 4 1.a even 1 1 trivial
1200.3.bg.h 4 5.b even 2 1 inner
1200.3.bg.h 4 5.c odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 144$$ acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$144 + T^{4}$$
$11$ $$( 6 + T )^{4}$$
$13$ $$90000 + T^{4}$$
$17$ $$186624 + T^{4}$$
$19$ $$( 100 + T^{2} )^{2}$$
$23$ $$2985984 + T^{4}$$
$29$ $$( 2304 + T^{2} )^{2}$$
$31$ $$( -26 + T )^{4}$$
$37$ $$4112784 + T^{4}$$
$41$ $$( -30 + T )^{4}$$
$43$ $$2985984 + T^{4}$$
$47$ $$186624 + T^{4}$$
$53$ $$186624 + T^{4}$$
$59$ $$( 6084 + T^{2} )^{2}$$
$61$ $$( -2 + T )^{4}$$
$67$ $$65804544 + T^{4}$$
$71$ $$( 120 + T )^{4}$$
$73$ $$192432384 + T^{4}$$
$79$ $$( 5476 + T^{2} )^{2}$$
$83$ $$15116544 + T^{4}$$
$89$ $$( 22500 + T^{2} )^{2}$$
$97$ $$2304 + T^{4}$$