# Properties

 Label 1200.3.c.g 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: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 600) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{7} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{7} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{11} + 15 \zeta_{8}^{2} q^{13} + ( 14 \zeta_{8} - 14 \zeta_{8}^{3} ) q^{17} -23 q^{19} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} + ( -10 \zeta_{8} + 23 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{27} + ( -18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{29} -33 q^{31} + ( 6 \zeta_{8} + 24 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{33} -66 \zeta_{8}^{2} q^{37} + ( -15 - 30 \zeta_{8} - 30 \zeta_{8}^{3} ) q^{39} + ( 26 \zeta_{8} + 26 \zeta_{8}^{3} ) q^{41} + 7 \zeta_{8}^{2} q^{43} + ( 32 \zeta_{8} - 32 \zeta_{8}^{3} ) q^{47} + 48 q^{49} + ( -56 + 14 \zeta_{8} + 14 \zeta_{8}^{3} ) q^{51} + ( -26 \zeta_{8} + 26 \zeta_{8}^{3} ) q^{53} + ( 46 \zeta_{8} - 23 \zeta_{8}^{2} - 46 \zeta_{8}^{3} ) q^{57} + ( -72 \zeta_{8} - 72 \zeta_{8}^{3} ) q^{59} + 39 q^{61} + ( 4 \zeta_{8} + 7 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} + 113 \zeta_{8}^{2} q^{67} + ( -8 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{69} + ( -18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{71} + 58 \zeta_{8}^{2} q^{73} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{77} + 70 q^{79} + ( 17 - 56 \zeta_{8} - 56 \zeta_{8}^{3} ) q^{81} + ( 108 \zeta_{8} - 108 \zeta_{8}^{3} ) q^{83} + ( 18 \zeta_{8} + 72 \zeta_{8}^{2} - 18 \zeta_{8}^{3} ) q^{87} + ( -64 \zeta_{8} - 64 \zeta_{8}^{3} ) q^{89} -15 q^{91} + ( 66 \zeta_{8} - 33 \zeta_{8}^{2} - 66 \zeta_{8}^{3} ) q^{93} + \zeta_{8}^{2} q^{97} + ( -48 - 42 \zeta_{8} - 42 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 28q^{9} + O(q^{10})$$ $$4q + 28q^{9} - 92q^{19} - 4q^{21} - 132q^{31} - 60q^{39} + 192q^{49} - 224q^{51} + 156q^{61} - 32q^{69} + 280q^{79} + 68q^{81} - 60q^{91} - 192q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 −2.82843 1.00000i 0 0 0 1.00000i 0 7.00000 + 5.65685i 0
449.2 0 −2.82843 + 1.00000i 0 0 0 1.00000i 0 7.00000 5.65685i 0
449.3 0 2.82843 1.00000i 0 0 0 1.00000i 0 7.00000 5.65685i 0
449.4 0 2.82843 + 1.00000i 0 0 0 1.00000i 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
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.g 4
3.b odd 2 1 inner 1200.3.c.g 4
4.b odd 2 1 600.3.c.b 4
5.b even 2 1 inner 1200.3.c.g 4
5.c odd 4 1 1200.3.l.j 2
5.c odd 4 1 1200.3.l.o 2
12.b even 2 1 600.3.c.b 4
15.d odd 2 1 inner 1200.3.c.g 4
15.e even 4 1 1200.3.l.j 2
15.e even 4 1 1200.3.l.o 2
20.d odd 2 1 600.3.c.b 4
20.e even 4 1 600.3.l.a 2
20.e even 4 1 600.3.l.c yes 2
60.h even 2 1 600.3.c.b 4
60.l odd 4 1 600.3.l.a 2
60.l odd 4 1 600.3.l.c yes 2

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

 $$T_{7}^{2} + 1$$ $$T_{11}^{2} + 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 14 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 72 + T^{2} )^{2}$$
$13$ $$( 225 + T^{2} )^{2}$$
$17$ $$( -392 + T^{2} )^{2}$$
$19$ $$( 23 + T )^{4}$$
$23$ $$( -8 + T^{2} )^{2}$$
$29$ $$( 648 + T^{2} )^{2}$$
$31$ $$( 33 + T )^{4}$$
$37$ $$( 4356 + T^{2} )^{2}$$
$41$ $$( 1352 + T^{2} )^{2}$$
$43$ $$( 49 + T^{2} )^{2}$$
$47$ $$( -2048 + T^{2} )^{2}$$
$53$ $$( -1352 + T^{2} )^{2}$$
$59$ $$( 10368 + T^{2} )^{2}$$
$61$ $$( -39 + T )^{4}$$
$67$ $$( 12769 + T^{2} )^{2}$$
$71$ $$( 648 + T^{2} )^{2}$$
$73$ $$( 3364 + T^{2} )^{2}$$
$79$ $$( -70 + T )^{4}$$
$83$ $$( -23328 + T^{2} )^{2}$$
$89$ $$( 8192 + T^{2} )^{2}$$
$97$ $$( 1 + T^{2} )^{2}$$