# Properties

 Label 1200.3.l.f 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: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-11})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 + \beta ) q^{3} + ( 6 - 5 \beta ) q^{9} +O(q^{10})$$ $$q + ( -3 + \beta ) q^{3} + ( 6 - 5 \beta ) q^{9} + ( 5 - 10 \beta ) q^{11} + 10 q^{13} + ( 1 - 2 \beta ) q^{17} -7 q^{19} + ( -6 + 12 \beta ) q^{23} + ( -3 + 16 \beta ) q^{27} + ( -10 + 20 \beta ) q^{29} -42 q^{31} + ( 15 + 25 \beta ) q^{33} -40 q^{37} + ( -30 + 10 \beta ) q^{39} + ( 5 - 10 \beta ) q^{41} + 50 q^{43} + ( 14 - 28 \beta ) q^{47} -49 q^{49} + ( 3 + 5 \beta ) q^{51} + ( -14 + 28 \beta ) q^{53} + ( 21 - 7 \beta ) q^{57} + ( 20 - 40 \beta ) q^{59} -8 q^{61} -45 q^{67} + ( -18 - 30 \beta ) q^{69} + ( 10 - 20 \beta ) q^{71} -35 q^{73} -12 q^{79} + ( -39 - 35 \beta ) q^{81} + ( -21 + 42 \beta ) q^{83} + ( -30 - 50 \beta ) q^{87} + ( 45 - 90 \beta ) q^{89} + ( 126 - 42 \beta ) q^{93} -70 q^{97} + ( -120 - 35 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 5q^{3} + 7q^{9} + O(q^{10})$$ $$2q - 5q^{3} + 7q^{9} + 20q^{13} - 14q^{19} + 10q^{27} - 84q^{31} + 55q^{33} - 80q^{37} - 50q^{39} + 100q^{43} - 98q^{49} + 11q^{51} + 35q^{57} - 16q^{61} - 90q^{67} - 66q^{69} - 70q^{73} - 24q^{79} - 113q^{81} - 110q^{87} + 210q^{93} - 140q^{97} - 275q^{99} + O(q^{100})$$

## 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
 0.5 − 1.65831i 0.5 + 1.65831i
0 −2.50000 1.65831i 0 0 0 0 0 3.50000 + 8.29156i 0
401.2 0 −2.50000 + 1.65831i 0 0 0 0 0 3.50000 8.29156i 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.f 2
3.b odd 2 1 inner 1200.3.l.f 2
4.b odd 2 1 75.3.c.f yes 2
5.b even 2 1 1200.3.l.s 2
5.c odd 4 2 1200.3.c.d 4
12.b even 2 1 75.3.c.f yes 2
15.d odd 2 1 1200.3.l.s 2
15.e even 4 2 1200.3.c.d 4
20.d odd 2 1 75.3.c.c 2
20.e even 4 2 75.3.d.c 4
60.h even 2 1 75.3.c.c 2
60.l odd 4 2 75.3.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.c 2 20.d odd 2 1
75.3.c.c 2 60.h even 2 1
75.3.c.f yes 2 4.b odd 2 1
75.3.c.f yes 2 12.b even 2 1
75.3.d.c 4 20.e even 4 2
75.3.d.c 4 60.l odd 4 2
1200.3.c.d 4 5.c odd 4 2
1200.3.c.d 4 15.e even 4 2
1200.3.l.f 2 1.a even 1 1 trivial
1200.3.l.f 2 3.b odd 2 1 inner
1200.3.l.s 2 5.b even 2 1
1200.3.l.s 2 15.d odd 2 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}^{2} + 275$$ $$T_{13} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 5 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$275 + T^{2}$$
$13$ $$( -10 + T )^{2}$$
$17$ $$11 + T^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$396 + T^{2}$$
$29$ $$1100 + T^{2}$$
$31$ $$( 42 + T )^{2}$$
$37$ $$( 40 + T )^{2}$$
$41$ $$275 + T^{2}$$
$43$ $$( -50 + T )^{2}$$
$47$ $$2156 + T^{2}$$
$53$ $$2156 + T^{2}$$
$59$ $$4400 + T^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$( 45 + T )^{2}$$
$71$ $$1100 + T^{2}$$
$73$ $$( 35 + T )^{2}$$
$79$ $$( 12 + T )^{2}$$
$83$ $$4851 + T^{2}$$
$89$ $$22275 + T^{2}$$
$97$ $$( 70 + T )^{2}$$