# Properties

 Label 1200.2.f.e Level $1200$ Weight $2$ Character orbit 1200.f Analytic conductor $9.582$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 4 i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + 4 i q^{7} - q^{9} -2 i q^{13} + 6 i q^{17} -4 q^{19} -4 q^{21} -i q^{27} + 6 q^{29} -8 q^{31} + 2 i q^{37} + 2 q^{39} -6 q^{41} -4 i q^{43} -9 q^{49} -6 q^{51} + 6 i q^{53} -4 i q^{57} -10 q^{61} -4 i q^{63} + 4 i q^{67} -2 i q^{73} + 8 q^{79} + q^{81} + 12 i q^{83} + 6 i q^{87} -18 q^{89} + 8 q^{91} -8 i q^{93} + 2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 8q^{19} - 8q^{21} + 12q^{29} - 16q^{31} + 4q^{39} - 12q^{41} - 18q^{49} - 12q^{51} - 20q^{61} + 16q^{79} + 2q^{81} - 36q^{89} + 16q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 4.00000i 0 −1.00000 0
49.2 0 1.00000i 0 0 0 4.00000i 0 −1.00000 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
5.b 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.f.e 2
3.b odd 2 1 3600.2.f.i 2
4.b odd 2 1 150.2.c.a 2
5.b even 2 1 inner 1200.2.f.e 2
5.c odd 4 1 240.2.a.b 1
5.c odd 4 1 1200.2.a.k 1
8.b even 2 1 4800.2.f.w 2
8.d odd 2 1 4800.2.f.p 2
12.b even 2 1 450.2.c.b 2
15.d odd 2 1 3600.2.f.i 2
15.e even 4 1 720.2.a.j 1
15.e even 4 1 3600.2.a.f 1
20.d odd 2 1 150.2.c.a 2
20.e even 4 1 30.2.a.a 1
20.e even 4 1 150.2.a.b 1
40.e odd 2 1 4800.2.f.p 2
40.f even 2 1 4800.2.f.w 2
40.i odd 4 1 960.2.a.p 1
40.i odd 4 1 4800.2.a.d 1
40.k even 4 1 960.2.a.e 1
40.k even 4 1 4800.2.a.cq 1
60.h even 2 1 450.2.c.b 2
60.l odd 4 1 90.2.a.c 1
60.l odd 4 1 450.2.a.d 1
80.i odd 4 1 3840.2.k.f 2
80.j even 4 1 3840.2.k.y 2
80.s even 4 1 3840.2.k.y 2
80.t odd 4 1 3840.2.k.f 2
120.q odd 4 1 2880.2.a.a 1
120.w even 4 1 2880.2.a.q 1
140.j odd 4 1 1470.2.a.d 1
140.j odd 4 1 7350.2.a.ct 1
140.w even 12 2 1470.2.i.o 2
140.x odd 12 2 1470.2.i.q 2
180.v odd 12 2 810.2.e.b 2
180.x even 12 2 810.2.e.l 2
220.i odd 4 1 3630.2.a.w 1
260.l odd 4 1 5070.2.b.k 2
260.p even 4 1 5070.2.a.w 1
260.s odd 4 1 5070.2.b.k 2
340.r even 4 1 8670.2.a.g 1
420.w even 4 1 4410.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 20.e even 4 1
90.2.a.c 1 60.l odd 4 1
150.2.a.b 1 20.e even 4 1
150.2.c.a 2 4.b odd 2 1
150.2.c.a 2 20.d odd 2 1
240.2.a.b 1 5.c odd 4 1
450.2.a.d 1 60.l odd 4 1
450.2.c.b 2 12.b even 2 1
450.2.c.b 2 60.h even 2 1
720.2.a.j 1 15.e even 4 1
810.2.e.b 2 180.v odd 12 2
810.2.e.l 2 180.x even 12 2
960.2.a.e 1 40.k even 4 1
960.2.a.p 1 40.i odd 4 1
1200.2.a.k 1 5.c odd 4 1
1200.2.f.e 2 1.a even 1 1 trivial
1200.2.f.e 2 5.b even 2 1 inner
1470.2.a.d 1 140.j odd 4 1
1470.2.i.o 2 140.w even 12 2
1470.2.i.q 2 140.x odd 12 2
2880.2.a.a 1 120.q odd 4 1
2880.2.a.q 1 120.w even 4 1
3600.2.a.f 1 15.e even 4 1
3600.2.f.i 2 3.b odd 2 1
3600.2.f.i 2 15.d odd 2 1
3630.2.a.w 1 220.i odd 4 1
3840.2.k.f 2 80.i odd 4 1
3840.2.k.f 2 80.t odd 4 1
3840.2.k.y 2 80.j even 4 1
3840.2.k.y 2 80.s even 4 1
4410.2.a.z 1 420.w even 4 1
4800.2.a.d 1 40.i odd 4 1
4800.2.a.cq 1 40.k even 4 1
4800.2.f.p 2 8.d odd 2 1
4800.2.f.p 2 40.e odd 2 1
4800.2.f.w 2 8.b even 2 1
4800.2.f.w 2 40.f even 2 1
5070.2.a.w 1 260.p even 4 1
5070.2.b.k 2 260.l odd 4 1
5070.2.b.k 2 260.s odd 4 1
7350.2.a.ct 1 140.j odd 4 1
8670.2.a.g 1 340.r even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ $$T_{11}$$ $$T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 18 + T )^{2}$$
$97$ $$4 + T^{2}$$