# Properties

 Label 1200.3.c.h 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 150) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{3} + 7 \zeta_{8}^{2} q^{7} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{3} + 7 \zeta_{8}^{2} q^{7} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{11} + 25 \zeta_{8}^{2} q^{13} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{17} -7 q^{19} + ( 7 + 14 \zeta_{8} + 14 \zeta_{8}^{3} ) q^{21} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{23} + ( 10 \zeta_{8} - 23 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{27} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{29} + 7 q^{31} + ( -6 \zeta_{8} - 24 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{33} + 2 \zeta_{8}^{2} q^{37} + ( 25 + 50 \zeta_{8} + 50 \zeta_{8}^{3} ) q^{39} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{41} + 41 \zeta_{8}^{2} q^{43} + ( 72 - 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{51} + ( 42 \zeta_{8} - 42 \zeta_{8}^{3} ) q^{53} + ( -14 \zeta_{8} + 7 \zeta_{8}^{2} + 14 \zeta_{8}^{3} ) q^{57} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{59} - q^{61} + ( 28 \zeta_{8} + 49 \zeta_{8}^{2} - 28 \zeta_{8}^{3} ) q^{63} -17 \zeta_{8}^{2} q^{67} + ( -72 + 18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{69} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{71} + 70 \zeta_{8}^{2} q^{73} + ( 42 \zeta_{8} - 42 \zeta_{8}^{3} ) q^{77} -58 q^{79} + ( 17 - 56 \zeta_{8} - 56 \zeta_{8}^{3} ) q^{81} + ( 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{83} + ( 30 \zeta_{8} + 120 \zeta_{8}^{2} - 30 \zeta_{8}^{3} ) q^{87} + ( 96 \zeta_{8} + 96 \zeta_{8}^{3} ) q^{89} -175 q^{91} + ( 14 \zeta_{8} - 7 \zeta_{8}^{2} - 14 \zeta_{8}^{3} ) q^{93} -49 \zeta_{8}^{2} q^{97} + ( -48 - 42 \zeta_{8} - 42 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 28q^{9} + O(q^{10})$$ $$4q + 28q^{9} - 28q^{19} + 28q^{21} + 28q^{31} + 100q^{39} + 288q^{51} - 4q^{61} - 288q^{69} - 232q^{79} + 68q^{81} - 700q^{91} - 192q^{99} + O(q^{100})$$

## 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 7.00000i 0 7.00000 + 5.65685i 0
449.2 0 −2.82843 + 1.00000i 0 0 0 7.00000i 0 7.00000 5.65685i 0
449.3 0 2.82843 1.00000i 0 0 0 7.00000i 0 7.00000 5.65685i 0
449.4 0 2.82843 + 1.00000i 0 0 0 7.00000i 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.h 4
3.b odd 2 1 inner 1200.3.c.h 4
4.b odd 2 1 150.3.b.a 4
5.b even 2 1 inner 1200.3.c.h 4
5.c odd 4 1 1200.3.l.i 2
5.c odd 4 1 1200.3.l.p 2
12.b even 2 1 150.3.b.a 4
15.d odd 2 1 inner 1200.3.c.h 4
15.e even 4 1 1200.3.l.i 2
15.e even 4 1 1200.3.l.p 2
20.d odd 2 1 150.3.b.a 4
20.e even 4 1 150.3.d.a 2
20.e even 4 1 150.3.d.b yes 2
60.h even 2 1 150.3.b.a 4
60.l odd 4 1 150.3.d.a 2
60.l odd 4 1 150.3.d.b yes 2

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

 $$T_{7}^{2} + 49$$ $$T_{11}^{2} + 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 14 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 49 + T^{2} )^{2}$$
$11$ $$( 72 + T^{2} )^{2}$$
$13$ $$( 625 + T^{2} )^{2}$$
$17$ $$( -648 + T^{2} )^{2}$$
$19$ $$( 7 + T )^{4}$$
$23$ $$( -648 + T^{2} )^{2}$$
$29$ $$( 1800 + T^{2} )^{2}$$
$31$ $$( -7 + T )^{4}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 72 + T^{2} )^{2}$$
$43$ $$( 1681 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( -3528 + T^{2} )^{2}$$
$59$ $$( 1152 + T^{2} )^{2}$$
$61$ $$( 1 + T )^{4}$$
$67$ $$( 289 + T^{2} )^{2}$$
$71$ $$( 1800 + T^{2} )^{2}$$
$73$ $$( 4900 + T^{2} )^{2}$$
$79$ $$( 58 + T )^{4}$$
$83$ $$( -14112 + T^{2} )^{2}$$
$89$ $$( 18432 + T^{2} )^{2}$$
$97$ $$( 2401 + T^{2} )^{2}$$