# Properties

 Label 1200.3.e.n 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3$$ Twist minimal: no (minimal twist has level 240) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{7} -3 q^{9} + ( 2 \beta_{1} - \beta_{2} ) q^{11} + ( 8 + \beta_{3} ) q^{13} + ( -12 - \beta_{3} ) q^{17} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 6 + \beta_{3} ) q^{21} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{23} + 3 \beta_{1} q^{27} + ( -12 + \beta_{3} ) q^{29} + ( 20 \beta_{1} - 2 \beta_{2} ) q^{31} + ( 6 - \beta_{3} ) q^{33} + ( -20 + 3 \beta_{3} ) q^{37} + ( -8 \beta_{1} - 3 \beta_{2} ) q^{39} -30 q^{41} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 32 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -23 - 4 \beta_{3} ) q^{49} + ( 12 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -24 - 3 \beta_{3} ) q^{53} + ( -12 - 2 \beta_{3} ) q^{57} + ( 2 \beta_{1} + 11 \beta_{2} ) q^{59} + ( 14 - 6 \beta_{3} ) q^{61} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{63} + ( 48 \beta_{1} - 6 \beta_{2} ) q^{67} + ( 12 + 4 \beta_{3} ) q^{69} + ( 68 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 2 + 2 \beta_{3} ) q^{73} + 48 q^{77} + ( -44 \beta_{1} + 2 \beta_{2} ) q^{79} + 9 q^{81} + ( 24 \beta_{1} - 6 \beta_{2} ) q^{83} + ( 12 \beta_{1} - 3 \beta_{2} ) q^{87} + ( 6 + 2 \beta_{3} ) q^{89} + ( 76 \beta_{1} + 14 \beta_{2} ) q^{91} + ( 60 - 2 \beta_{3} ) q^{93} + 14 q^{97} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9} + O(q^{10})$$ $$4 q - 12 q^{9} + 32 q^{13} - 48 q^{17} + 24 q^{21} - 48 q^{29} + 24 q^{33} - 80 q^{37} - 120 q^{41} - 92 q^{49} - 96 q^{53} - 48 q^{57} + 56 q^{61} + 48 q^{69} + 8 q^{73} + 192 q^{77} + 36 q^{81} + 24 q^{89} + 240 q^{93} + 56 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{3} + 2 \nu^{2} - 2 \nu$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{3} - 4 \nu^{2} + 12 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$-6 \nu^{3} - 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{2} + 6 \beta_{1} + 6$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 3 \beta_{2} + 18 \beta_{1} - 18$$$$)/24$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} - 12$$$$)/6$$

## 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
 −0.309017 − 0.535233i 0.809017 + 1.40126i 0.809017 − 1.40126i −0.309017 + 0.535233i
0 1.73205i 0 0 0 4.28187i 0 −3.00000 0
751.2 0 1.73205i 0 0 0 11.2101i 0 −3.00000 0
751.3 0 1.73205i 0 0 0 11.2101i 0 −3.00000 0
751.4 0 1.73205i 0 0 0 4.28187i 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.n 4
3.b odd 2 1 3600.3.e.bh 4
4.b odd 2 1 inner 1200.3.e.n 4
5.b even 2 1 240.3.e.a 4
5.c odd 4 2 1200.3.j.f 8
12.b even 2 1 3600.3.e.bh 4
15.d odd 2 1 720.3.e.a 4
15.e even 4 2 3600.3.j.l 8
20.d odd 2 1 240.3.e.a 4
20.e even 4 2 1200.3.j.f 8
40.e odd 2 1 960.3.e.b 4
40.f even 2 1 960.3.e.b 4
60.h even 2 1 720.3.e.a 4
60.l odd 4 2 3600.3.j.l 8
120.i odd 2 1 2880.3.e.f 4
120.m even 2 1 2880.3.e.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.e.a 4 5.b even 2 1
240.3.e.a 4 20.d odd 2 1
720.3.e.a 4 15.d odd 2 1
720.3.e.a 4 60.h even 2 1
960.3.e.b 4 40.e odd 2 1
960.3.e.b 4 40.f even 2 1
1200.3.e.n 4 1.a even 1 1 trivial
1200.3.e.n 4 4.b odd 2 1 inner
1200.3.j.f 8 5.c odd 4 2
1200.3.j.f 8 20.e even 4 2
2880.3.e.f 4 120.i odd 2 1
2880.3.e.f 4 120.m even 2 1
3600.3.e.bh 4 3.b odd 2 1
3600.3.e.bh 4 12.b even 2 1
3600.3.j.l 8 15.e even 4 2
3600.3.j.l 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 $$S_{3}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{4} + 144 T_{7}^{2} + 2304$$ $$T_{11}^{4} + 144 T_{11}^{2} + 2304$$ $$T_{13}^{2} - 16 T_{13} - 116$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$2304 + 144 T^{2} + T^{4}$$
$11$ $$2304 + 144 T^{2} + T^{4}$$
$13$ $$( -116 - 16 T + T^{2} )^{2}$$
$17$ $$( -36 + 24 T + T^{2} )^{2}$$
$19$ $$36864 + 576 T^{2} + T^{4}$$
$23$ $$831744 + 2016 T^{2} + T^{4}$$
$29$ $$( -36 + 24 T + T^{2} )^{2}$$
$31$ $$921600 + 2880 T^{2} + T^{4}$$
$37$ $$( -1220 + 40 T + T^{2} )^{2}$$
$41$ $$( 30 + T )^{4}$$
$43$ $$831744 + 2016 T^{2} + T^{4}$$
$47$ $$8020224 + 6624 T^{2} + T^{4}$$
$53$ $$( -1044 + 48 T + T^{2} )^{2}$$
$59$ $$52533504 + 14544 T^{2} + T^{4}$$
$61$ $$( -6284 - 28 T + T^{2} )^{2}$$
$67$ $$22581504 + 18144 T^{2} + T^{4}$$
$71$ $$185831424 + 28224 T^{2} + T^{4}$$
$73$ $$( -716 - 4 T + T^{2} )^{2}$$
$79$ $$31002624 + 12096 T^{2} + T^{4}$$
$83$ $$186624 + 7776 T^{2} + T^{4}$$
$89$ $$( -684 - 12 T + T^{2} )^{2}$$
$97$ $$( -14 + T )^{4}$$