# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{3} ) q^{3} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{3} ) q^{3} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{9} + 2 \beta_{1} q^{11} + 10 q^{13} + ( -4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -8 + 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -13 + 6 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{21} + ( \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{23} + ( 7 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{27} + ( 4 \beta_{2} - 8 \beta_{3} ) q^{29} -8 q^{31} + ( -6 + 6 \beta_{2} ) q^{33} + ( 22 + 4 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{37} + ( 10 + 10 \beta_{3} ) q^{39} + ( -8 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 14 + 3 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{43} + ( 5 \beta_{1} - 10 \beta_{2} + 20 \beta_{3} ) q^{47} + ( 45 + 4 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{49} + ( -18 - 6 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} ) q^{51} + ( -4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{53} + ( -38 + 12 \beta_{1} - 6 \beta_{2} - 8 \beta_{3} ) q^{57} + ( -12 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{59} + ( -16 + 4 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{61} + ( -64 - \beta_{1} + 14 \beta_{2} - 8 \beta_{3} ) q^{63} + ( -82 + 3 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{67} + ( -33 - 6 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{69} + ( 10 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{71} + ( -50 + 4 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{73} + ( 4 \beta_{1} - 12 \beta_{2} + 24 \beta_{3} ) q^{77} + ( 28 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 7 + 4 \beta_{1} - 20 \beta_{2} + 8 \beta_{3} ) q^{81} + ( -3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{83} + ( 60 + 12 \beta_{1} - 12 \beta_{3} ) q^{87} + ( -8 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{89} + ( 20 + 10 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} ) q^{91} + ( -8 - 8 \beta_{3} ) q^{93} + ( -74 - 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{97} + ( 48 - 6 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 8 q^{7} - 8 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} + 8 q^{7} - 8 q^{9} + 40 q^{13} - 32 q^{19} - 52 q^{21} + 28 q^{27} - 32 q^{31} - 24 q^{33} + 88 q^{37} + 40 q^{39} + 56 q^{43} + 180 q^{49} - 72 q^{51} - 152 q^{57} - 64 q^{61} - 256 q^{63} - 328 q^{67} - 132 q^{69} - 200 q^{73} + 112 q^{79} + 28 q^{81} + 240 q^{87} + 80 q^{91} - 32 q^{93} - 296 q^{97} + 192 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2 \nu - 2$$ $$\beta_{3}$$ $$=$$ $$-\nu^{2} + \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{3} + \beta_{2} + 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 3 \beta_{1}$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 −1.58114 − 0.707107i −1.58114 + 0.707107i 1.58114 + 0.707107i 1.58114 − 0.707107i
0 −0.581139 2.94317i 0 0 0 11.4868 0 −8.32456 + 3.42079i 0
401.2 0 −0.581139 + 2.94317i 0 0 0 11.4868 0 −8.32456 3.42079i 0
401.3 0 2.58114 1.52896i 0 0 0 −7.48683 0 4.32456 7.89292i 0
401.4 0 2.58114 + 1.52896i 0 0 0 −7.48683 0 4.32456 + 7.89292i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.l.u 4
3.b odd 2 1 inner 1200.3.l.u 4
4.b odd 2 1 150.3.d.c 4
5.b even 2 1 240.3.l.c 4
5.c odd 4 2 1200.3.c.k 8
12.b even 2 1 150.3.d.c 4
15.d odd 2 1 240.3.l.c 4
15.e even 4 2 1200.3.c.k 8
20.d odd 2 1 30.3.d.a 4
20.e even 4 2 150.3.b.b 8
40.e odd 2 1 960.3.l.e 4
40.f even 2 1 960.3.l.f 4
60.h even 2 1 30.3.d.a 4
60.l odd 4 2 150.3.b.b 8
120.i odd 2 1 960.3.l.f 4
120.m even 2 1 960.3.l.e 4
180.n even 6 2 810.3.h.a 8
180.p odd 6 2 810.3.h.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 20.d odd 2 1
30.3.d.a 4 60.h even 2 1
150.3.b.b 8 20.e even 4 2
150.3.b.b 8 60.l odd 4 2
150.3.d.c 4 4.b odd 2 1
150.3.d.c 4 12.b even 2 1
240.3.l.c 4 5.b even 2 1
240.3.l.c 4 15.d odd 2 1
810.3.h.a 8 180.n even 6 2
810.3.h.a 8 180.p odd 6 2
960.3.l.e 4 40.e odd 2 1
960.3.l.e 4 120.m even 2 1
960.3.l.f 4 40.f even 2 1
960.3.l.f 4 120.i odd 2 1
1200.3.c.k 8 5.c odd 4 2
1200.3.c.k 8 15.e even 4 2
1200.3.l.u 4 1.a even 1 1 trivial
1200.3.l.u 4 3.b odd 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} - 4 T_{7} - 86$$ $$T_{11}^{2} + 72$$ $$T_{13} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 36 T + 12 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -86 - 4 T + T^{2} )^{2}$$
$11$ $$( 72 + T^{2} )^{2}$$
$13$ $$( -10 + T )^{4}$$
$17$ $$11664 + 936 T^{2} + T^{4}$$
$19$ $$( -296 + 16 T + T^{2} )^{2}$$
$23$ $$26244 + 396 T^{2} + T^{4}$$
$29$ $$( 720 + T^{2} )^{2}$$
$31$ $$( 8 + T )^{4}$$
$37$ $$( -956 - 44 T + T^{2} )^{2}$$
$41$ $$944784 + 2664 T^{2} + T^{4}$$
$43$ $$( -614 - 28 T + T^{2} )^{2}$$
$47$ $$16402500 + 9900 T^{2} + T^{4}$$
$53$ $$11664 + 936 T^{2} + T^{4}$$
$59$ $$3504384 + 6624 T^{2} + T^{4}$$
$61$ $$( -1184 + 32 T + T^{2} )^{2}$$
$67$ $$( 5914 + 164 T + T^{2} )^{2}$$
$71$ $$1166400 + 5040 T^{2} + T^{4}$$
$73$ $$( 1060 + 100 T + T^{2} )^{2}$$
$79$ $$( 424 - 56 T + T^{2} )^{2}$$
$83$ $$324 + 684 T^{2} + T^{4}$$
$89$ $$186624 + 3744 T^{2} + T^{4}$$
$97$ $$( 4036 + 148 T + T^{2} )^{2}$$