# Properties

 Label 1200.3.bg.o 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 60) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 5 - 5 \beta_{2} - 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 5 - 5 \beta_{2} - 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( 4 + 5 \beta_{1} - 5 \beta_{3} ) q^{11} -10 \beta_{1} q^{13} + ( 10 - 10 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -10 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{19} + ( -6 - 5 \beta_{1} + 5 \beta_{3} ) q^{21} + ( -10 + 6 \beta_{1} - 10 \beta_{2} ) q^{23} -3 \beta_{3} q^{27} + ( 5 \beta_{1} + 22 \beta_{2} + 5 \beta_{3} ) q^{29} + ( -4 + 10 \beta_{1} - 10 \beta_{3} ) q^{31} + ( -15 - 4 \beta_{1} - 15 \beta_{2} ) q^{33} + 6 \beta_{3} q^{37} + 30 \beta_{2} q^{39} + ( 50 - 10 \beta_{1} + 10 \beta_{3} ) q^{41} + ( 30 - 4 \beta_{1} + 30 \beta_{2} ) q^{43} -18 \beta_{3} q^{47} + ( -20 \beta_{1} - 13 \beta_{2} - 20 \beta_{3} ) q^{49} + ( 6 - 10 \beta_{1} + 10 \beta_{3} ) q^{51} + ( 50 - 12 \beta_{1} + 50 \beta_{2} ) q^{53} + ( -30 + 30 \beta_{2} - 10 \beta_{3} ) q^{57} + ( -15 \beta_{1} - 52 \beta_{2} - 15 \beta_{3} ) q^{59} + ( -78 - 10 \beta_{1} + 10 \beta_{3} ) q^{61} + ( 15 + 6 \beta_{1} + 15 \beta_{2} ) q^{63} + ( -10 + 10 \beta_{2} + 12 \beta_{3} ) q^{67} + ( 10 \beta_{1} - 18 \beta_{2} + 10 \beta_{3} ) q^{69} + ( 20 - 40 \beta_{1} + 40 \beta_{3} ) q^{71} + ( 5 + 32 \beta_{1} + 5 \beta_{2} ) q^{73} + ( 50 - 50 \beta_{2} - 58 \beta_{3} ) q^{77} + ( -10 \beta_{1} - 24 \beta_{2} - 10 \beta_{3} ) q^{79} -9 q^{81} + ( -60 - 34 \beta_{1} - 60 \beta_{2} ) q^{83} + ( 15 - 15 \beta_{2} - 22 \beta_{3} ) q^{87} + ( -60 \beta_{1} - 10 \beta_{2} - 60 \beta_{3} ) q^{89} + ( -60 - 50 \beta_{1} + 50 \beta_{3} ) q^{91} + ( -30 + 4 \beta_{1} - 30 \beta_{2} ) q^{93} + ( -75 + 75 \beta_{2} - 16 \beta_{3} ) q^{97} + ( 15 \beta_{1} + 12 \beta_{2} + 15 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 20q^{7} + O(q^{10})$$ $$4q + 20q^{7} + 16q^{11} + 40q^{17} - 24q^{21} - 40q^{23} - 16q^{31} - 60q^{33} + 200q^{41} + 120q^{43} + 24q^{51} + 200q^{53} - 120q^{57} - 312q^{61} + 60q^{63} - 40q^{67} + 80q^{71} + 20q^{73} + 200q^{77} - 36q^{81} - 240q^{83} + 60q^{87} - 240q^{91} - 120q^{93} - 300q^{97} + O(q^{100})$$

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 7.44949 7.44949i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 2.55051 2.55051i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 7.44949 + 7.44949i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 2.55051 + 2.55051i 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.o 4
4.b odd 2 1 300.3.k.a 4
5.b even 2 1 240.3.bg.d 4
5.c odd 4 1 240.3.bg.d 4
5.c odd 4 1 inner 1200.3.bg.o 4
12.b even 2 1 900.3.l.b 4
15.d odd 2 1 720.3.bh.f 4
15.e even 4 1 720.3.bh.f 4
20.d odd 2 1 60.3.k.a 4
20.e even 4 1 60.3.k.a 4
20.e even 4 1 300.3.k.a 4
40.e odd 2 1 960.3.bg.b 4
40.f even 2 1 960.3.bg.a 4
40.i odd 4 1 960.3.bg.a 4
40.k even 4 1 960.3.bg.b 4
60.h even 2 1 180.3.l.b 4
60.l odd 4 1 180.3.l.b 4
60.l odd 4 1 900.3.l.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.k.a 4 20.d odd 2 1
60.3.k.a 4 20.e even 4 1
180.3.l.b 4 60.h even 2 1
180.3.l.b 4 60.l odd 4 1
240.3.bg.d 4 5.b even 2 1
240.3.bg.d 4 5.c odd 4 1
300.3.k.a 4 4.b odd 2 1
300.3.k.a 4 20.e even 4 1
720.3.bh.f 4 15.d odd 2 1
720.3.bh.f 4 15.e even 4 1
900.3.l.b 4 12.b even 2 1
900.3.l.b 4 60.l odd 4 1
960.3.bg.a 4 40.f even 2 1
960.3.bg.a 4 40.i odd 4 1
960.3.bg.b 4 40.e odd 2 1
960.3.bg.b 4 40.k even 4 1
1200.3.bg.o 4 1.a even 1 1 trivial
1200.3.bg.o 4 5.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 20 T_{7}^{3} + 200 T_{7}^{2} - 760 T_{7} + 1444$$ acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$1444 - 760 T + 200 T^{2} - 20 T^{3} + T^{4}$$
$11$ $$( -134 - 8 T + T^{2} )^{2}$$
$13$ $$90000 + T^{4}$$
$17$ $$35344 - 7520 T + 800 T^{2} - 40 T^{3} + T^{4}$$
$19$ $$250000 + 1400 T^{2} + T^{4}$$
$23$ $$8464 + 3680 T + 800 T^{2} + 40 T^{3} + T^{4}$$
$29$ $$111556 + 1268 T^{2} + T^{4}$$
$31$ $$( -584 + 8 T + T^{2} )^{2}$$
$37$ $$11664 + T^{4}$$
$41$ $$( 1900 - 100 T + T^{2} )^{2}$$
$43$ $$3069504 - 210240 T + 7200 T^{2} - 120 T^{3} + T^{4}$$
$47$ $$944784 + T^{4}$$
$53$ $$20866624 - 913600 T + 20000 T^{2} - 200 T^{3} + T^{4}$$
$59$ $$1833316 + 8108 T^{2} + T^{4}$$
$61$ $$( 5484 + 156 T + T^{2} )^{2}$$
$67$ $$53824 - 9280 T + 800 T^{2} + 40 T^{3} + T^{4}$$
$71$ $$( -9200 - 40 T + T^{2} )^{2}$$
$73$ $$9132484 + 60440 T + 200 T^{2} - 20 T^{3} + T^{4}$$
$79$ $$576 + 2352 T^{2} + T^{4}$$
$83$ $$13927824 + 895680 T + 28800 T^{2} + 240 T^{3} + T^{4}$$
$89$ $$462250000 + 43400 T^{2} + T^{4}$$
$97$ $$109872324 + 3144600 T + 45000 T^{2} + 300 T^{3} + T^{4}$$