# Properties

 Label 1200.3.e.m Level $1200$ Weight $3$ Character orbit 1200.e 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{11})$$ Defining polynomial: $$x^{4} + 11 x^{2} + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( \beta_{1} + \beta_{2} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( \beta_{1} + \beta_{2} ) q^{7} -3 q^{9} + \beta_{1} q^{11} + q^{13} + ( -12 - \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{2} ) q^{19} + ( -3 - \beta_{3} ) q^{21} + ( \beta_{1} + 12 \beta_{2} ) q^{23} -3 \beta_{2} q^{27} + ( -24 + \beta_{3} ) q^{29} + ( -\beta_{1} - 15 \beta_{2} ) q^{31} -\beta_{3} q^{33} + ( -22 + 2 \beta_{3} ) q^{37} + \beta_{2} q^{39} + ( -6 + 3 \beta_{3} ) q^{41} + ( -\beta_{1} + 21 \beta_{2} ) q^{43} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{47} + ( -86 - 2 \beta_{3} ) q^{49} + ( -3 \beta_{1} - 12 \beta_{2} ) q^{51} + ( 30 + 3 \beta_{3} ) q^{53} + ( 3 + \beta_{3} ) q^{57} + ( 4 \beta_{1} - 30 \beta_{2} ) q^{59} + ( -37 - 2 \beta_{3} ) q^{61} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{63} + ( -3 \beta_{1} - 43 \beta_{2} ) q^{67} + ( -36 - \beta_{3} ) q^{69} + ( \beta_{1} - 6 \beta_{2} ) q^{71} + ( 70 - 2 \beta_{3} ) q^{73} + ( -132 - \beta_{3} ) q^{77} + ( 4 \beta_{1} - 28 \beta_{2} ) q^{79} + 9 q^{81} + 42 \beta_{2} q^{83} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{87} -96 q^{89} + ( \beta_{1} + \beta_{2} ) q^{91} + ( 45 + \beta_{3} ) q^{93} + ( -35 - 4 \beta_{3} ) q^{97} -3 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9} + O(q^{10})$$ $$4 q - 12 q^{9} + 4 q^{13} - 48 q^{17} - 12 q^{21} - 96 q^{29} - 88 q^{37} - 24 q^{41} - 344 q^{49} + 120 q^{53} + 12 q^{57} - 148 q^{61} - 144 q^{69} + 280 q^{73} - 528 q^{77} + 36 q^{81} - 384 q^{89} + 180 q^{93} - 140 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11 x^{2} + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} + 44 \nu$$$$)/11$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{2} + 11$$$$)/11$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{3}$$$$)/11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$11 \beta_{2} - 11$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-11 \beta_{3}$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 1.65831 − 2.87228i −1.65831 + 2.87228i −1.65831 − 2.87228i 1.65831 + 2.87228i
0 1.73205i 0 0 0 13.2212i 0 −3.00000 0
751.2 0 1.73205i 0 0 0 9.75707i 0 −3.00000 0
751.3 0 1.73205i 0 0 0 9.75707i 0 −3.00000 0
751.4 0 1.73205i 0 0 0 13.2212i 0 −3.00000 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
4.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.e.m yes 4
3.b odd 2 1 3600.3.e.bf 4
4.b odd 2 1 inner 1200.3.e.m yes 4
5.b even 2 1 1200.3.e.k 4
5.c odd 4 2 1200.3.j.d 8
12.b even 2 1 3600.3.e.bf 4
15.d odd 2 1 3600.3.e.bc 4
15.e even 4 2 3600.3.j.j 8
20.d odd 2 1 1200.3.e.k 4
20.e even 4 2 1200.3.j.d 8
60.h even 2 1 3600.3.e.bc 4
60.l odd 4 2 3600.3.j.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.3.e.k 4 5.b even 2 1
1200.3.e.k 4 20.d odd 2 1
1200.3.e.m yes 4 1.a even 1 1 trivial
1200.3.e.m yes 4 4.b odd 2 1 inner
1200.3.j.d 8 5.c odd 4 2
1200.3.j.d 8 20.e even 4 2
3600.3.e.bc 4 15.d odd 2 1
3600.3.e.bc 4 60.h even 2 1
3600.3.e.bf 4 3.b odd 2 1
3600.3.e.bf 4 12.b even 2 1
3600.3.j.j 8 15.e even 4 2
3600.3.j.j 8 60.l odd 4 2

## Hecke kernels

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

 $$T_{7}^{4} + 270 T_{7}^{2} + 16641$$ $$T_{11}^{2} + 132$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$16641 + 270 T^{2} + T^{4}$$
$11$ $$( 132 + T^{2} )^{2}$$
$13$ $$( -1 + T )^{4}$$
$17$ $$( -252 + 24 T + T^{2} )^{2}$$
$19$ $$16641 + 270 T^{2} + T^{4}$$
$23$ $$90000 + 1128 T^{2} + T^{4}$$
$29$ $$( 180 + 48 T + T^{2} )^{2}$$
$31$ $$294849 + 1614 T^{2} + T^{4}$$
$37$ $$( -1100 + 44 T + T^{2} )^{2}$$
$41$ $$( -3528 + 12 T + T^{2} )^{2}$$
$43$ $$1418481 + 2910 T^{2} + T^{4}$$
$47$ $$4016016 + 4440 T^{2} + T^{4}$$
$53$ $$( -2664 - 60 T + T^{2} )^{2}$$
$59$ $$345744 + 9624 T^{2} + T^{4}$$
$61$ $$( -215 + 74 T + T^{2} )^{2}$$
$67$ $$19000881 + 13470 T^{2} + T^{4}$$
$71$ $$576 + 480 T^{2} + T^{4}$$
$73$ $$( 3316 - 140 T + T^{2} )^{2}$$
$79$ $$57600 + 8928 T^{2} + T^{4}$$
$83$ $$( 5292 + T^{2} )^{2}$$
$89$ $$( 96 + T )^{4}$$
$97$ $$( -5111 + 70 T + T^{2} )^{2}$$