# Properties

 Label 1200.4.f.r Level $1200$ Weight $4$ Character orbit 1200.f Analytic conductor $70.802$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$70.8022920069$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 4 i q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + 4 i q^{7} -9 q^{9} + 48 q^{11} -2 i q^{13} -114 i q^{17} + 140 q^{19} -12 q^{21} + 72 i q^{23} -27 i q^{27} -210 q^{29} -272 q^{31} + 144 i q^{33} -334 i q^{37} + 6 q^{39} -198 q^{41} -268 i q^{43} -216 i q^{47} + 327 q^{49} + 342 q^{51} + 78 i q^{53} + 420 i q^{57} + 240 q^{59} + 302 q^{61} -36 i q^{63} -596 i q^{67} -216 q^{69} + 768 q^{71} + 478 i q^{73} + 192 i q^{77} -640 q^{79} + 81 q^{81} -348 i q^{83} -630 i q^{87} -210 q^{89} + 8 q^{91} -816 i q^{93} -1534 i q^{97} -432 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9} + O(q^{10})$$ $$2 q - 18 q^{9} + 96 q^{11} + 280 q^{19} - 24 q^{21} - 420 q^{29} - 544 q^{31} + 12 q^{39} - 396 q^{41} + 654 q^{49} + 684 q^{51} + 480 q^{59} + 604 q^{61} - 432 q^{69} + 1536 q^{71} - 1280 q^{79} + 162 q^{81} - 420 q^{89} + 16 q^{91} - 864 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 4.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 4.00000i 0 −9.00000 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 1200.4.f.r 2
4.b odd 2 1 150.4.c.c 2
5.b even 2 1 inner 1200.4.f.r 2
5.c odd 4 1 240.4.a.b 1
5.c odd 4 1 1200.4.a.ba 1
12.b even 2 1 450.4.c.j 2
15.e even 4 1 720.4.a.y 1
20.d odd 2 1 150.4.c.c 2
20.e even 4 1 30.4.a.b 1
20.e even 4 1 150.4.a.b 1
40.i odd 4 1 960.4.a.bg 1
40.k even 4 1 960.4.a.n 1
60.h even 2 1 450.4.c.j 2
60.l odd 4 1 90.4.a.c 1
60.l odd 4 1 450.4.a.r 1
140.j odd 4 1 1470.4.a.r 1
180.v odd 12 2 810.4.e.p 2
180.x even 12 2 810.4.e.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.b 1 20.e even 4 1
90.4.a.c 1 60.l odd 4 1
150.4.a.b 1 20.e even 4 1
150.4.c.c 2 4.b odd 2 1
150.4.c.c 2 20.d odd 2 1
240.4.a.b 1 5.c odd 4 1
450.4.a.r 1 60.l odd 4 1
450.4.c.j 2 12.b even 2 1
450.4.c.j 2 60.h even 2 1
720.4.a.y 1 15.e even 4 1
810.4.e.i 2 180.x even 12 2
810.4.e.p 2 180.v odd 12 2
960.4.a.n 1 40.k even 4 1
960.4.a.bg 1 40.i odd 4 1
1200.4.a.ba 1 5.c odd 4 1
1200.4.f.r 2 1.a even 1 1 trivial
1200.4.f.r 2 5.b even 2 1 inner
1470.4.a.r 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 16$$ $$T_{11} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( -48 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$12996 + T^{2}$$
$19$ $$( -140 + T )^{2}$$
$23$ $$5184 + T^{2}$$
$29$ $$( 210 + T )^{2}$$
$31$ $$( 272 + T )^{2}$$
$37$ $$111556 + T^{2}$$
$41$ $$( 198 + T )^{2}$$
$43$ $$71824 + T^{2}$$
$47$ $$46656 + T^{2}$$
$53$ $$6084 + T^{2}$$
$59$ $$( -240 + T )^{2}$$
$61$ $$( -302 + T )^{2}$$
$67$ $$355216 + T^{2}$$
$71$ $$( -768 + T )^{2}$$
$73$ $$228484 + T^{2}$$
$79$ $$( 640 + T )^{2}$$
$83$ $$121104 + T^{2}$$
$89$ $$( 210 + T )^{2}$$
$97$ $$2353156 + T^{2}$$