# Properties

 Label 1200.4.f.b 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 24 i q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + 24 i q^{7} -9 q^{9} -52 q^{11} -22 i q^{13} -14 i q^{17} -20 q^{19} -72 q^{21} -168 i q^{23} -27 i q^{27} -230 q^{29} + 288 q^{31} -156 i q^{33} -34 i q^{37} + 66 q^{39} + 122 q^{41} -188 i q^{43} -256 i q^{47} -233 q^{49} + 42 q^{51} + 338 i q^{53} -60 i q^{57} + 100 q^{59} + 742 q^{61} -216 i q^{63} + 84 i q^{67} + 504 q^{69} + 328 q^{71} + 38 i q^{73} -1248 i q^{77} -240 q^{79} + 81 q^{81} + 1212 i q^{83} -690 i q^{87} -330 q^{89} + 528 q^{91} + 864 i q^{93} + 866 i q^{97} + 468 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9} + O(q^{10})$$ $$2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81} - 660 q^{89} + 1056 q^{91} + 936 q^{99} + O(q^{100})$$

## 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 24.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 24.0000i 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.b 2
4.b odd 2 1 75.4.b.b 2
5.b even 2 1 inner 1200.4.f.b 2
5.c odd 4 1 240.4.a.e 1
5.c odd 4 1 1200.4.a.t 1
12.b even 2 1 225.4.b.e 2
15.e even 4 1 720.4.a.n 1
20.d odd 2 1 75.4.b.b 2
20.e even 4 1 15.4.a.a 1
20.e even 4 1 75.4.a.b 1
40.i odd 4 1 960.4.a.ba 1
40.k even 4 1 960.4.a.b 1
60.h even 2 1 225.4.b.e 2
60.l odd 4 1 45.4.a.c 1
60.l odd 4 1 225.4.a.f 1
140.j odd 4 1 735.4.a.e 1
180.v odd 12 2 405.4.e.i 2
180.x even 12 2 405.4.e.g 2
220.i odd 4 1 1815.4.a.e 1
420.w even 4 1 2205.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 20.e even 4 1
45.4.a.c 1 60.l odd 4 1
75.4.a.b 1 20.e even 4 1
75.4.b.b 2 4.b odd 2 1
75.4.b.b 2 20.d odd 2 1
225.4.a.f 1 60.l odd 4 1
225.4.b.e 2 12.b even 2 1
225.4.b.e 2 60.h even 2 1
240.4.a.e 1 5.c odd 4 1
405.4.e.g 2 180.x even 12 2
405.4.e.i 2 180.v odd 12 2
720.4.a.n 1 15.e even 4 1
735.4.a.e 1 140.j odd 4 1
960.4.a.b 1 40.k even 4 1
960.4.a.ba 1 40.i odd 4 1
1200.4.a.t 1 5.c odd 4 1
1200.4.f.b 2 1.a even 1 1 trivial
1200.4.f.b 2 5.b even 2 1 inner
1815.4.a.e 1 220.i odd 4 1
2205.4.a.l 1 420.w even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 576$$ $$T_{11} + 52$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$576 + T^{2}$$
$11$ $$( 52 + T )^{2}$$
$13$ $$484 + T^{2}$$
$17$ $$196 + T^{2}$$
$19$ $$( 20 + T )^{2}$$
$23$ $$28224 + T^{2}$$
$29$ $$( 230 + T )^{2}$$
$31$ $$( -288 + T )^{2}$$
$37$ $$1156 + T^{2}$$
$41$ $$( -122 + T )^{2}$$
$43$ $$35344 + T^{2}$$
$47$ $$65536 + T^{2}$$
$53$ $$114244 + T^{2}$$
$59$ $$( -100 + T )^{2}$$
$61$ $$( -742 + T )^{2}$$
$67$ $$7056 + T^{2}$$
$71$ $$( -328 + T )^{2}$$
$73$ $$1444 + T^{2}$$
$79$ $$( 240 + T )^{2}$$
$83$ $$1468944 + T^{2}$$
$89$ $$( 330 + T )^{2}$$
$97$ $$749956 + T^{2}$$