# Properties

 Label 1200.2.o.f Level $1200$ Weight $2$ Character orbit 1200.o 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.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 240) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( \beta_{1} - \beta_{3} ) q^{7} -3 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( \beta_{1} - \beta_{3} ) q^{7} -3 \beta_{2} q^{9} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{11} + 2 \beta_{2} q^{13} + 6 q^{17} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{19} + ( -3 + 3 \beta_{2} ) q^{21} + ( -\beta_{1} - \beta_{3} ) q^{23} + 3 \beta_{1} q^{27} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{31} + ( 6 - 6 \beta_{2} ) q^{33} + 2 \beta_{2} q^{37} -2 \beta_{1} q^{39} -6 \beta_{2} q^{41} + ( \beta_{1} - \beta_{3} ) q^{43} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{47} - q^{49} + 6 \beta_{3} q^{51} -6 q^{53} + ( -6 - 6 \beta_{2} ) q^{57} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{59} + 8 q^{61} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{63} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{67} + ( 3 + 3 \beta_{2} ) q^{69} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{71} + 14 \beta_{2} q^{73} -12 q^{77} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{79} -9 q^{81} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{83} + 12 \beta_{2} q^{89} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{91} + ( -12 - 12 \beta_{2} ) q^{93} + 10 \beta_{2} q^{97} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 24q^{17} - 12q^{21} + 24q^{33} - 4q^{49} - 24q^{53} - 24q^{57} + 32q^{61} + 12q^{69} - 48q^{77} - 36q^{81} - 48q^{93} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1199.1
 1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 − 1.22474i −1.22474 + 1.22474i
0 −1.22474 1.22474i 0 0 0 2.44949 0 3.00000i 0
1199.2 0 −1.22474 + 1.22474i 0 0 0 2.44949 0 3.00000i 0
1199.3 0 1.22474 1.22474i 0 0 0 −2.44949 0 3.00000i 0
1199.4 0 1.22474 + 1.22474i 0 0 0 −2.44949 0 3.00000i 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
4.b odd 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.o.f 4
3.b odd 2 1 1200.2.o.e 4
4.b odd 2 1 inner 1200.2.o.f 4
5.b even 2 1 1200.2.o.e 4
5.c odd 4 1 240.2.h.a 4
5.c odd 4 1 1200.2.h.k 4
12.b even 2 1 1200.2.o.e 4
15.d odd 2 1 inner 1200.2.o.f 4
15.e even 4 1 240.2.h.a 4
15.e even 4 1 1200.2.h.k 4
20.d odd 2 1 1200.2.o.e 4
20.e even 4 1 240.2.h.a 4
20.e even 4 1 1200.2.h.k 4
40.i odd 4 1 960.2.h.c 4
40.k even 4 1 960.2.h.c 4
60.h even 2 1 inner 1200.2.o.f 4
60.l odd 4 1 240.2.h.a 4
60.l odd 4 1 1200.2.h.k 4
120.q odd 4 1 960.2.h.c 4
120.w even 4 1 960.2.h.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.h.a 4 5.c odd 4 1
240.2.h.a 4 15.e even 4 1
240.2.h.a 4 20.e even 4 1
240.2.h.a 4 60.l odd 4 1
960.2.h.c 4 40.i odd 4 1
960.2.h.c 4 40.k even 4 1
960.2.h.c 4 120.q odd 4 1
960.2.h.c 4 120.w even 4 1
1200.2.h.k 4 5.c odd 4 1
1200.2.h.k 4 15.e even 4 1
1200.2.h.k 4 20.e even 4 1
1200.2.h.k 4 60.l odd 4 1
1200.2.o.e 4 3.b odd 2 1
1200.2.o.e 4 5.b even 2 1
1200.2.o.e 4 12.b even 2 1
1200.2.o.e 4 20.d odd 2 1
1200.2.o.f 4 1.a even 1 1 trivial
1200.2.o.f 4 4.b odd 2 1 inner
1200.2.o.f 4 15.d odd 2 1 inner
1200.2.o.f 4 60.h even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} - 6$$ $$T_{11}^{2} - 24$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -6 + T^{2} )^{2}$$
$11$ $$( -24 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( -6 + T )^{4}$$
$19$ $$( 24 + T^{2} )^{2}$$
$23$ $$( 6 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 96 + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 36 + T^{2} )^{2}$$
$43$ $$( -6 + T^{2} )^{2}$$
$47$ $$( 150 + T^{2} )^{2}$$
$53$ $$( 6 + T )^{4}$$
$59$ $$( -96 + T^{2} )^{2}$$
$61$ $$( -8 + T )^{4}$$
$67$ $$( -54 + T^{2} )^{2}$$
$71$ $$( -24 + T^{2} )^{2}$$
$73$ $$( 196 + T^{2} )^{2}$$
$79$ $$( 24 + T^{2} )^{2}$$
$83$ $$( 54 + T^{2} )^{2}$$
$89$ $$( 144 + T^{2} )^{2}$$
$97$ $$( 100 + T^{2} )^{2}$$