# Properties

 Label 1200.3.c.e Level $1200$ Weight $3$ Character orbit 1200.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{3} ) q^{3} + \beta_{1} q^{7} + ( 1 + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{3} ) q^{3} + \beta_{1} q^{7} + ( 1 + 2 \beta_{2} ) q^{9} -3 \beta_{2} q^{11} + 4 \beta_{1} q^{13} + 6 \beta_{3} q^{17} -34 q^{19} + ( 4 - \beta_{2} ) q^{21} + 18 \beta_{3} q^{23} + ( -11 \beta_{1} + 7 \beta_{3} ) q^{27} + 9 \beta_{2} q^{29} -14 q^{31} + ( 15 \beta_{1} - 12 \beta_{3} ) q^{33} -28 \beta_{1} q^{37} + ( 16 - 4 \beta_{2} ) q^{39} -6 \beta_{2} q^{41} -4 \beta_{1} q^{43} + 18 \beta_{3} q^{47} + 45 q^{49} + ( -30 - 6 \beta_{2} ) q^{51} + 18 \beta_{3} q^{53} + ( 34 \beta_{1} + 34 \beta_{3} ) q^{57} + 3 \beta_{2} q^{59} -46 q^{61} + ( \beta_{1} - 8 \beta_{3} ) q^{63} + 16 \beta_{1} q^{67} + ( -90 - 18 \beta_{2} ) q^{69} -12 \beta_{2} q^{71} -53 \beta_{1} q^{73} + 12 \beta_{3} q^{77} -22 q^{79} + ( -79 + 4 \beta_{2} ) q^{81} -54 \beta_{3} q^{83} + ( -45 \beta_{1} + 36 \beta_{3} ) q^{87} -24 \beta_{2} q^{89} -16 q^{91} + ( 14 \beta_{1} + 14 \beta_{3} ) q^{93} -61 \beta_{1} q^{97} + ( 120 - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} - 136q^{19} + 16q^{21} - 56q^{31} + 64q^{39} + 180q^{49} - 120q^{51} - 184q^{61} - 360q^{69} - 88q^{79} - 316q^{81} - 64q^{91} + 480q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 2 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.618034i − 0.618034i − 1.61803i 1.61803i
0 −2.23607 2.00000i 0 0 0 2.00000i 0 1.00000 + 8.94427i 0
449.2 0 −2.23607 + 2.00000i 0 0 0 2.00000i 0 1.00000 8.94427i 0
449.3 0 2.23607 2.00000i 0 0 0 2.00000i 0 1.00000 8.94427i 0
449.4 0 2.23607 + 2.00000i 0 0 0 2.00000i 0 1.00000 + 8.94427i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.c.e 4
3.b odd 2 1 inner 1200.3.c.e 4
4.b odd 2 1 300.3.b.c 4
5.b even 2 1 inner 1200.3.c.e 4
5.c odd 4 1 240.3.l.a 2
5.c odd 4 1 1200.3.l.r 2
12.b even 2 1 300.3.b.c 4
15.d odd 2 1 inner 1200.3.c.e 4
15.e even 4 1 240.3.l.a 2
15.e even 4 1 1200.3.l.r 2
20.d odd 2 1 300.3.b.c 4
20.e even 4 1 60.3.g.a 2
20.e even 4 1 300.3.g.d 2
40.i odd 4 1 960.3.l.d 2
40.k even 4 1 960.3.l.a 2
60.h even 2 1 300.3.b.c 4
60.l odd 4 1 60.3.g.a 2
60.l odd 4 1 300.3.g.d 2
120.q odd 4 1 960.3.l.a 2
120.w even 4 1 960.3.l.d 2
180.v odd 12 2 1620.3.o.b 4
180.x even 12 2 1620.3.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 20.e even 4 1
60.3.g.a 2 60.l odd 4 1
240.3.l.a 2 5.c odd 4 1
240.3.l.a 2 15.e even 4 1
300.3.b.c 4 4.b odd 2 1
300.3.b.c 4 12.b even 2 1
300.3.b.c 4 20.d odd 2 1
300.3.b.c 4 60.h even 2 1
300.3.g.d 2 20.e even 4 1
300.3.g.d 2 60.l odd 4 1
960.3.l.a 2 40.k even 4 1
960.3.l.a 2 120.q odd 4 1
960.3.l.d 2 40.i odd 4 1
960.3.l.d 2 120.w even 4 1
1200.3.c.e 4 1.a even 1 1 trivial
1200.3.c.e 4 3.b odd 2 1 inner
1200.3.c.e 4 5.b even 2 1 inner
1200.3.c.e 4 15.d odd 2 1 inner
1200.3.l.r 2 5.c odd 4 1
1200.3.l.r 2 15.e even 4 1
1620.3.o.b 4 180.v odd 12 2
1620.3.o.b 4 180.x even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11}^{2} + 180$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 2 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( 180 + T^{2} )^{2}$$
$13$ $$( 64 + T^{2} )^{2}$$
$17$ $$( -180 + T^{2} )^{2}$$
$19$ $$( 34 + T )^{4}$$
$23$ $$( -1620 + T^{2} )^{2}$$
$29$ $$( 1620 + T^{2} )^{2}$$
$31$ $$( 14 + T )^{4}$$
$37$ $$( 3136 + T^{2} )^{2}$$
$41$ $$( 720 + T^{2} )^{2}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$( -1620 + T^{2} )^{2}$$
$53$ $$( -1620 + T^{2} )^{2}$$
$59$ $$( 180 + T^{2} )^{2}$$
$61$ $$( 46 + T )^{4}$$
$67$ $$( 1024 + T^{2} )^{2}$$
$71$ $$( 2880 + T^{2} )^{2}$$
$73$ $$( 11236 + T^{2} )^{2}$$
$79$ $$( 22 + T )^{4}$$
$83$ $$( -14580 + T^{2} )^{2}$$
$89$ $$( 11520 + T^{2} )^{2}$$
$97$ $$( 14884 + T^{2} )^{2}$$