# Properties

 Label 1200.3.l.n Level $1200$ Weight $3$ Character orbit 1200.l Analytic conductor $32.698$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} -6 q^{7} + ( -7 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} -6 q^{7} + ( -7 + 2 \beta ) q^{9} + 2 \beta q^{11} -10 q^{13} -8 \beta q^{17} -2 q^{19} + ( -6 - 6 \beta ) q^{21} + 4 \beta q^{23} + ( -23 - 5 \beta ) q^{27} -6 \beta q^{29} + 22 q^{31} + ( -16 + 2 \beta ) q^{33} + 6 q^{37} + ( -10 - 10 \beta ) q^{39} -12 \beta q^{41} + 82 q^{43} -24 \beta q^{47} -13 q^{49} + ( 64 - 8 \beta ) q^{51} -22 \beta q^{53} + ( -2 - 2 \beta ) q^{57} + 26 \beta q^{59} -86 q^{61} + ( 42 - 12 \beta ) q^{63} + 2 q^{67} + ( -32 + 4 \beta ) q^{69} -44 \beta q^{71} -82 q^{73} -12 \beta q^{77} -10 q^{79} + ( 17 - 28 \beta ) q^{81} + 26 \beta q^{83} + ( 48 - 6 \beta ) q^{87} + 12 \beta q^{89} + 60 q^{91} + ( 22 + 22 \beta ) q^{93} + 94 q^{97} + ( -32 - 14 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 12q^{7} - 14q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 12q^{7} - 14q^{9} - 20q^{13} - 4q^{19} - 12q^{21} - 46q^{27} + 44q^{31} - 32q^{33} + 12q^{37} - 20q^{39} + 164q^{43} - 26q^{49} + 128q^{51} - 4q^{57} - 172q^{61} + 84q^{63} + 4q^{67} - 64q^{69} - 164q^{73} - 20q^{79} + 34q^{81} + 96q^{87} + 120q^{91} + 44q^{93} + 188q^{97} - 64q^{99} + O(q^{100})$$

## 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
 − 1.41421i 1.41421i
0 1.00000 2.82843i 0 0 0 −6.00000 0 −7.00000 5.65685i 0
401.2 0 1.00000 + 2.82843i 0 0 0 −6.00000 0 −7.00000 + 5.65685i 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.n 2
3.b odd 2 1 inner 1200.3.l.n 2
4.b odd 2 1 600.3.l.b 2
5.b even 2 1 48.3.e.b 2
5.c odd 4 2 1200.3.c.i 4
12.b even 2 1 600.3.l.b 2
15.d odd 2 1 48.3.e.b 2
15.e even 4 2 1200.3.c.i 4
20.d odd 2 1 24.3.e.a 2
20.e even 4 2 600.3.c.a 4
40.e odd 2 1 192.3.e.c 2
40.f even 2 1 192.3.e.d 2
45.h odd 6 2 1296.3.q.e 4
45.j even 6 2 1296.3.q.e 4
60.h even 2 1 24.3.e.a 2
60.l odd 4 2 600.3.c.a 4
80.k odd 4 2 768.3.h.d 4
80.q even 4 2 768.3.h.c 4
120.i odd 2 1 192.3.e.d 2
120.m even 2 1 192.3.e.c 2
140.c even 2 1 1176.3.d.a 2
180.n even 6 2 648.3.m.d 4
180.p odd 6 2 648.3.m.d 4
240.t even 4 2 768.3.h.d 4
240.bm odd 4 2 768.3.h.c 4
420.o odd 2 1 1176.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 20.d odd 2 1
24.3.e.a 2 60.h even 2 1
48.3.e.b 2 5.b even 2 1
48.3.e.b 2 15.d odd 2 1
192.3.e.c 2 40.e odd 2 1
192.3.e.c 2 120.m even 2 1
192.3.e.d 2 40.f even 2 1
192.3.e.d 2 120.i odd 2 1
600.3.c.a 4 20.e even 4 2
600.3.c.a 4 60.l odd 4 2
600.3.l.b 2 4.b odd 2 1
600.3.l.b 2 12.b even 2 1
648.3.m.d 4 180.n even 6 2
648.3.m.d 4 180.p odd 6 2
768.3.h.c 4 80.q even 4 2
768.3.h.c 4 240.bm odd 4 2
768.3.h.d 4 80.k odd 4 2
768.3.h.d 4 240.t even 4 2
1176.3.d.a 2 140.c even 2 1
1176.3.d.a 2 420.o odd 2 1
1200.3.c.i 4 5.c odd 4 2
1200.3.c.i 4 15.e even 4 2
1200.3.l.n 2 1.a even 1 1 trivial
1200.3.l.n 2 3.b odd 2 1 inner
1296.3.q.e 4 45.h odd 6 2
1296.3.q.e 4 45.j even 6 2

## Hecke kernels

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

 $$T_{7} + 6$$ $$T_{11}^{2} + 32$$ $$T_{13} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 6 + T )^{2}$$
$11$ $$32 + T^{2}$$
$13$ $$( 10 + T )^{2}$$
$17$ $$512 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$128 + T^{2}$$
$29$ $$288 + T^{2}$$
$31$ $$( -22 + T )^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$1152 + T^{2}$$
$43$ $$( -82 + T )^{2}$$
$47$ $$4608 + T^{2}$$
$53$ $$3872 + T^{2}$$
$59$ $$5408 + T^{2}$$
$61$ $$( 86 + T )^{2}$$
$67$ $$( -2 + T )^{2}$$
$71$ $$15488 + T^{2}$$
$73$ $$( 82 + T )^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$5408 + T^{2}$$
$89$ $$1152 + T^{2}$$
$97$ $$( -94 + T )^{2}$$