# Properties

 Label 1200.3.c.d 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: $$\Q(i, \sqrt{11})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$5^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -3 + \beta_{3} ) q^{9} + ( -1 - 2 \beta_{3} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( \beta_{1} - \beta_{2} ) q^{17} + 7 q^{19} + ( 6 \beta_{1} - 6 \beta_{2} ) q^{23} + ( -7 \beta_{1} + 9 \beta_{2} ) q^{27} + ( -2 - 4 \beta_{3} ) q^{29} -42 q^{31} + ( 7 \beta_{1} - 18 \beta_{2} ) q^{33} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{37} + ( 24 - 2 \beta_{3} ) q^{39} + ( -1 - 2 \beta_{3} ) q^{41} + ( -10 \beta_{1} - 10 \beta_{2} ) q^{43} + ( 14 \beta_{1} - 14 \beta_{2} ) q^{47} + 49 q^{49} + ( 6 + \beta_{3} ) q^{51} + ( 14 \beta_{1} - 14 \beta_{2} ) q^{53} + 7 \beta_{1} q^{57} + ( 4 + 8 \beta_{3} ) q^{59} -8 q^{61} + ( -9 \beta_{1} - 9 \beta_{2} ) q^{67} + ( 36 + 6 \beta_{3} ) q^{69} + ( -2 - 4 \beta_{3} ) q^{71} + ( 7 \beta_{1} + 7 \beta_{2} ) q^{73} + 12 q^{79} + ( -60 - 7 \beta_{3} ) q^{81} + ( 21 \beta_{1} - 21 \beta_{2} ) q^{83} + ( 14 \beta_{1} - 36 \beta_{2} ) q^{87} + ( 9 + 18 \beta_{3} ) q^{89} -42 \beta_{1} q^{93} + ( -14 \beta_{1} - 14 \beta_{2} ) q^{97} + ( 141 + 7 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 14q^{9} + O(q^{10})$$ $$4q - 14q^{9} + 28q^{19} - 168q^{31} + 100q^{39} + 196q^{49} + 22q^{51} - 32q^{61} + 132q^{69} + 48q^{79} - 226q^{81} + 550q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} + \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 3 \nu$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{2} - 13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{2} - 3 \beta_{1}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 13$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} - 9 \beta_{1}$$$$)/5$$

## 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
 1.65831 + 0.500000i 1.65831 − 0.500000i −1.65831 + 0.500000i −1.65831 − 0.500000i
0 −1.65831 2.50000i 0 0 0 0 0 −3.50000 + 8.29156i 0
449.2 0 −1.65831 + 2.50000i 0 0 0 0 0 −3.50000 8.29156i 0
449.3 0 1.65831 2.50000i 0 0 0 0 0 −3.50000 8.29156i 0
449.4 0 1.65831 + 2.50000i 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
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.d 4
3.b odd 2 1 inner 1200.3.c.d 4
4.b odd 2 1 75.3.d.c 4
5.b even 2 1 inner 1200.3.c.d 4
5.c odd 4 1 1200.3.l.f 2
5.c odd 4 1 1200.3.l.s 2
12.b even 2 1 75.3.d.c 4
15.d odd 2 1 inner 1200.3.c.d 4
15.e even 4 1 1200.3.l.f 2
15.e even 4 1 1200.3.l.s 2
20.d odd 2 1 75.3.d.c 4
20.e even 4 1 75.3.c.c 2
20.e even 4 1 75.3.c.f yes 2
60.h even 2 1 75.3.d.c 4
60.l odd 4 1 75.3.c.c 2
60.l odd 4 1 75.3.c.f yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.c 2 20.e even 4 1
75.3.c.c 2 60.l odd 4 1
75.3.c.f yes 2 20.e even 4 1
75.3.c.f yes 2 60.l odd 4 1
75.3.d.c 4 4.b odd 2 1
75.3.d.c 4 12.b even 2 1
75.3.d.c 4 20.d odd 2 1
75.3.d.c 4 60.h even 2 1
1200.3.c.d 4 1.a even 1 1 trivial
1200.3.c.d 4 3.b odd 2 1 inner
1200.3.c.d 4 5.b even 2 1 inner
1200.3.c.d 4 15.d odd 2 1 inner
1200.3.l.f 2 5.c odd 4 1
1200.3.l.f 2 15.e even 4 1
1200.3.l.s 2 5.c odd 4 1
1200.3.l.s 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 $$S_{3}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}$$ $$T_{11}^{2} + 275$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 + 7 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 275 + T^{2} )^{2}$$
$13$ $$( 100 + T^{2} )^{2}$$
$17$ $$( -11 + T^{2} )^{2}$$
$19$ $$( -7 + T )^{4}$$
$23$ $$( -396 + T^{2} )^{2}$$
$29$ $$( 1100 + T^{2} )^{2}$$
$31$ $$( 42 + T )^{4}$$
$37$ $$( 1600 + T^{2} )^{2}$$
$41$ $$( 275 + T^{2} )^{2}$$
$43$ $$( 2500 + T^{2} )^{2}$$
$47$ $$( -2156 + T^{2} )^{2}$$
$53$ $$( -2156 + T^{2} )^{2}$$
$59$ $$( 4400 + T^{2} )^{2}$$
$61$ $$( 8 + T )^{4}$$
$67$ $$( 2025 + T^{2} )^{2}$$
$71$ $$( 1100 + T^{2} )^{2}$$
$73$ $$( 1225 + T^{2} )^{2}$$
$79$ $$( -12 + T )^{4}$$
$83$ $$( -4851 + T^{2} )^{2}$$
$89$ $$( 22275 + T^{2} )^{2}$$
$97$ $$( 4900 + T^{2} )^{2}$$