# Properties

 Label 1200.3.c.f 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{3} -3 \beta_{1} q^{7} + ( 1 + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{3} -3 \beta_{1} q^{7} + ( 1 + 2 \beta_{2} ) q^{9} -\beta_{2} q^{11} + 8 \beta_{1} q^{13} + 2 \beta_{3} q^{17} -2 q^{19} + ( 12 - 3 \beta_{2} ) q^{21} + 6 \beta_{3} q^{23} + ( 11 \beta_{1} - 7 \beta_{3} ) q^{27} + 7 \beta_{2} q^{29} + 18 q^{31} + ( -5 \beta_{1} + 4 \beta_{3} ) q^{33} + 8 \beta_{1} q^{37} + ( -32 + 8 \beta_{2} ) q^{39} + 14 \beta_{2} q^{41} -8 \beta_{1} q^{43} + 22 \beta_{3} q^{47} + 13 q^{49} + ( 10 + 2 \beta_{2} ) q^{51} -2 \beta_{3} q^{53} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{57} + \beta_{2} q^{59} + 82 q^{61} + ( -3 \beta_{1} + 24 \beta_{3} ) q^{63} + 12 \beta_{1} q^{67} + ( 30 + 6 \beta_{2} ) q^{69} + 28 \beta_{2} q^{71} -37 \beta_{1} q^{73} -12 \beta_{3} q^{77} + 138 q^{79} + ( -79 + 4 \beta_{2} ) q^{81} -42 \beta_{3} q^{83} + ( 35 \beta_{1} - 28 \beta_{3} ) q^{87} -24 \beta_{2} q^{89} + 96 q^{91} + ( 18 \beta_{1} + 18 \beta_{3} ) q^{93} + 83 \beta_{1} q^{97} + ( 40 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} - 8q^{19} + 48q^{21} + 72q^{31} - 128q^{39} + 52q^{49} + 40q^{51} + 328q^{61} + 120q^{69} + 552q^{79} - 316q^{81} + 384q^{91} + 160q^{99} + O(q^{100})$$

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
 1.61803i − 1.61803i − 0.618034i 0.618034i
0 −2.23607 2.00000i 0 0 0 6.00000i 0 1.00000 + 8.94427i 0
449.2 0 −2.23607 + 2.00000i 0 0 0 6.00000i 0 1.00000 8.94427i 0
449.3 0 2.23607 2.00000i 0 0 0 6.00000i 0 1.00000 8.94427i 0
449.4 0 2.23607 + 2.00000i 0 0 0 6.00000i 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.f 4
3.b odd 2 1 inner 1200.3.c.f 4
4.b odd 2 1 75.3.d.b 4
5.b even 2 1 inner 1200.3.c.f 4
5.c odd 4 1 240.3.l.b 2
5.c odd 4 1 1200.3.l.g 2
12.b even 2 1 75.3.d.b 4
15.d odd 2 1 inner 1200.3.c.f 4
15.e even 4 1 240.3.l.b 2
15.e even 4 1 1200.3.l.g 2
20.d odd 2 1 75.3.d.b 4
20.e even 4 1 15.3.c.a 2
20.e even 4 1 75.3.c.e 2
40.i odd 4 1 960.3.l.b 2
40.k even 4 1 960.3.l.c 2
60.h even 2 1 75.3.d.b 4
60.l odd 4 1 15.3.c.a 2
60.l odd 4 1 75.3.c.e 2
120.q odd 4 1 960.3.l.c 2
120.w even 4 1 960.3.l.b 2
180.v odd 12 2 405.3.i.b 4
180.x even 12 2 405.3.i.b 4

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

## Hecke kernels

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

 $$T_{7}^{2} + 36$$ $$T_{11}^{2} + 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 2 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 36 + T^{2} )^{2}$$
$11$ $$( 20 + T^{2} )^{2}$$
$13$ $$( 256 + T^{2} )^{2}$$
$17$ $$( -20 + T^{2} )^{2}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$( -180 + T^{2} )^{2}$$
$29$ $$( 980 + T^{2} )^{2}$$
$31$ $$( -18 + T )^{4}$$
$37$ $$( 256 + T^{2} )^{2}$$
$41$ $$( 3920 + T^{2} )^{2}$$
$43$ $$( 256 + T^{2} )^{2}$$
$47$ $$( -2420 + T^{2} )^{2}$$
$53$ $$( -20 + T^{2} )^{2}$$
$59$ $$( 20 + T^{2} )^{2}$$
$61$ $$( -82 + T )^{4}$$
$67$ $$( 576 + T^{2} )^{2}$$
$71$ $$( 15680 + T^{2} )^{2}$$
$73$ $$( 5476 + T^{2} )^{2}$$
$79$ $$( -138 + T )^{4}$$
$83$ $$( -8820 + T^{2} )^{2}$$
$89$ $$( 11520 + T^{2} )^{2}$$
$97$ $$( 27556 + T^{2} )^{2}$$