# Properties

 Label 1200.2.v.k Level $1200$ Weight $2$ Character orbit 1200.v Analytic conductor $9.582$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.v (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 - 2 \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 - 2 \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} -4 \zeta_{8} q^{17} -4 \zeta_{8}^{2} q^{19} + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} -6 \zeta_{8}^{3} q^{23} + ( 1 - 5 \zeta_{8} - \zeta_{8}^{2} ) q^{27} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} -8 q^{31} + ( 4 + 4 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{33} + ( 8 - 8 \zeta_{8}^{2} ) q^{37} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( 2 + 2 \zeta_{8}^{2} ) q^{43} -2 \zeta_{8} q^{47} -\zeta_{8}^{2} q^{49} + ( 4 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{51} + 8 \zeta_{8}^{3} q^{53} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{57} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{59} -6 q^{61} + ( 2 + 2 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{63} + ( 6 - 6 \zeta_{8}^{2} ) q^{67} + ( 6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{69} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{71} + ( 8 + 8 \zeta_{8}^{2} ) q^{73} -16 \zeta_{8} q^{77} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} -14 \zeta_{8}^{3} q^{83} + ( 4 - 8 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{87} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89} + ( -8 - 8 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{93} + ( -8 + 8 \zeta_{8}^{2} ) q^{97} + ( 4 \zeta_{8} + 16 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 8q^{7} + O(q^{10})$$ $$4q + 4q^{3} + 8q^{7} + 16q^{21} + 4q^{27} - 32q^{31} + 16q^{33} + 32q^{37} + 8q^{43} + 16q^{51} + 16q^{57} - 24q^{61} + 8q^{63} + 24q^{67} + 32q^{73} + 28q^{81} + 16q^{87} - 32q^{93} - 32q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}^{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0.292893 1.70711i 0 0 0 2.00000 + 2.00000i 0 −2.82843 1.00000i 0
257.2 0 1.70711 0.292893i 0 0 0 2.00000 + 2.00000i 0 2.82843 1.00000i 0
593.1 0 0.292893 + 1.70711i 0 0 0 2.00000 2.00000i 0 −2.82843 + 1.00000i 0
593.2 0 1.70711 + 0.292893i 0 0 0 2.00000 2.00000i 0 2.82843 + 1.00000i 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.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.v.k 4
3.b odd 2 1 inner 1200.2.v.k 4
4.b odd 2 1 600.2.r.a 4
5.b even 2 1 1200.2.v.a 4
5.c odd 4 1 1200.2.v.a 4
5.c odd 4 1 inner 1200.2.v.k 4
12.b even 2 1 600.2.r.a 4
15.d odd 2 1 1200.2.v.a 4
15.e even 4 1 1200.2.v.a 4
15.e even 4 1 inner 1200.2.v.k 4
20.d odd 2 1 600.2.r.e yes 4
20.e even 4 1 600.2.r.a 4
20.e even 4 1 600.2.r.e yes 4
60.h even 2 1 600.2.r.e yes 4
60.l odd 4 1 600.2.r.a 4
60.l odd 4 1 600.2.r.e yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.r.a 4 4.b odd 2 1
600.2.r.a 4 12.b even 2 1
600.2.r.a 4 20.e even 4 1
600.2.r.a 4 60.l odd 4 1
600.2.r.e yes 4 20.d odd 2 1
600.2.r.e yes 4 20.e even 4 1
600.2.r.e yes 4 60.h even 2 1
600.2.r.e yes 4 60.l odd 4 1
1200.2.v.a 4 5.b even 2 1
1200.2.v.a 4 5.c odd 4 1
1200.2.v.a 4 15.d odd 2 1
1200.2.v.a 4 15.e even 4 1
1200.2.v.k 4 1.a even 1 1 trivial
1200.2.v.k 4 3.b odd 2 1 inner
1200.2.v.k 4 5.c odd 4 1 inner
1200.2.v.k 4 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} - 4 T_{7} + 8$$ $$T_{11}^{2} + 32$$ $$T_{17}^{4} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 8 - 4 T + T^{2} )^{2}$$
$11$ $$( 32 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$256 + T^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$1296 + T^{4}$$
$29$ $$( -32 + T^{2} )^{2}$$
$31$ $$( 8 + T )^{4}$$
$37$ $$( 128 - 16 T + T^{2} )^{2}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$( 8 - 4 T + T^{2} )^{2}$$
$47$ $$16 + T^{4}$$
$53$ $$4096 + T^{4}$$
$59$ $$( -32 + T^{2} )^{2}$$
$61$ $$( 6 + T )^{4}$$
$67$ $$( 72 - 12 T + T^{2} )^{2}$$
$71$ $$( 128 + T^{2} )^{2}$$
$73$ $$( 128 - 16 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$38416 + T^{4}$$
$89$ $$( -128 + T^{2} )^{2}$$
$97$ $$( 128 + 16 T + T^{2} )^{2}$$