# Properties

 Label 1200.4.f.d 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 12) 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.

 $$f(q)$$ $$=$$ $$q -3 i q^{3} + 8 i q^{7} -9 q^{9} +O(q^{10})$$ $$q -3 i q^{3} + 8 i q^{7} -9 q^{9} -36 q^{11} -10 i q^{13} -18 i q^{17} -100 q^{19} + 24 q^{21} -72 i q^{23} + 27 i q^{27} + 234 q^{29} + 16 q^{31} + 108 i q^{33} + 226 i q^{37} -30 q^{39} + 90 q^{41} -452 i q^{43} + 432 i q^{47} + 279 q^{49} -54 q^{51} + 414 i q^{53} + 300 i q^{57} -684 q^{59} + 422 q^{61} -72 i q^{63} + 332 i q^{67} -216 q^{69} + 360 q^{71} + 26 i q^{73} -288 i q^{77} + 512 q^{79} + 81 q^{81} + 1188 i q^{83} -702 i q^{87} + 630 q^{89} + 80 q^{91} -48 i q^{93} + 1054 i q^{97} + 324 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9} + O(q^{10})$$ $$2 q - 18 q^{9} - 72 q^{11} - 200 q^{19} + 48 q^{21} + 468 q^{29} + 32 q^{31} - 60 q^{39} + 180 q^{41} + 558 q^{49} - 108 q^{51} - 1368 q^{59} + 844 q^{61} - 432 q^{69} + 720 q^{71} + 1024 q^{79} + 162 q^{81} + 1260 q^{89} + 160 q^{91} + 648 q^{99} + O(q^{100})$$

## 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 8.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 8.00000i 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.d 2
4.b odd 2 1 300.4.d.e 2
5.b even 2 1 inner 1200.4.f.d 2
5.c odd 4 1 48.4.a.a 1
5.c odd 4 1 1200.4.a.be 1
12.b even 2 1 900.4.d.c 2
15.e even 4 1 144.4.a.g 1
20.d odd 2 1 300.4.d.e 2
20.e even 4 1 12.4.a.a 1
20.e even 4 1 300.4.a.b 1
35.f even 4 1 2352.4.a.bk 1
40.i odd 4 1 192.4.a.l 1
40.k even 4 1 192.4.a.f 1
60.h even 2 1 900.4.d.c 2
60.l odd 4 1 36.4.a.a 1
60.l odd 4 1 900.4.a.g 1
80.i odd 4 1 768.4.d.j 2
80.j even 4 1 768.4.d.g 2
80.s even 4 1 768.4.d.g 2
80.t odd 4 1 768.4.d.j 2
120.q odd 4 1 576.4.a.b 1
120.w even 4 1 576.4.a.a 1
140.j odd 4 1 588.4.a.c 1
140.w even 12 2 588.4.i.d 2
140.x odd 12 2 588.4.i.e 2
180.v odd 12 2 324.4.e.a 2
180.x even 12 2 324.4.e.h 2
220.i odd 4 1 1452.4.a.d 1
260.l odd 4 1 2028.4.b.c 2
260.p even 4 1 2028.4.a.c 1
260.s odd 4 1 2028.4.b.c 2
420.w even 4 1 1764.4.a.b 1
420.bp odd 12 2 1764.4.k.b 2
420.br even 12 2 1764.4.k.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 20.e even 4 1
36.4.a.a 1 60.l odd 4 1
48.4.a.a 1 5.c odd 4 1
144.4.a.g 1 15.e even 4 1
192.4.a.f 1 40.k even 4 1
192.4.a.l 1 40.i odd 4 1
300.4.a.b 1 20.e even 4 1
300.4.d.e 2 4.b odd 2 1
300.4.d.e 2 20.d odd 2 1
324.4.e.a 2 180.v odd 12 2
324.4.e.h 2 180.x even 12 2
576.4.a.a 1 120.w even 4 1
576.4.a.b 1 120.q odd 4 1
588.4.a.c 1 140.j odd 4 1
588.4.i.d 2 140.w even 12 2
588.4.i.e 2 140.x odd 12 2
768.4.d.g 2 80.j even 4 1
768.4.d.g 2 80.s even 4 1
768.4.d.j 2 80.i odd 4 1
768.4.d.j 2 80.t odd 4 1
900.4.a.g 1 60.l odd 4 1
900.4.d.c 2 12.b even 2 1
900.4.d.c 2 60.h even 2 1
1200.4.a.be 1 5.c odd 4 1
1200.4.f.d 2 1.a even 1 1 trivial
1200.4.f.d 2 5.b even 2 1 inner
1452.4.a.d 1 220.i odd 4 1
1764.4.a.b 1 420.w even 4 1
1764.4.k.b 2 420.bp odd 12 2
1764.4.k.o 2 420.br even 12 2
2028.4.a.c 1 260.p even 4 1
2028.4.b.c 2 260.l odd 4 1
2028.4.b.c 2 260.s odd 4 1
2352.4.a.bk 1 35.f even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 64$$ $$T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$64 + T^{2}$$
$11$ $$( 36 + T )^{2}$$
$13$ $$100 + T^{2}$$
$17$ $$324 + T^{2}$$
$19$ $$( 100 + T )^{2}$$
$23$ $$5184 + T^{2}$$
$29$ $$( -234 + T )^{2}$$
$31$ $$( -16 + T )^{2}$$
$37$ $$51076 + T^{2}$$
$41$ $$( -90 + T )^{2}$$
$43$ $$204304 + T^{2}$$
$47$ $$186624 + T^{2}$$
$53$ $$171396 + T^{2}$$
$59$ $$( 684 + T )^{2}$$
$61$ $$( -422 + T )^{2}$$
$67$ $$110224 + T^{2}$$
$71$ $$( -360 + T )^{2}$$
$73$ $$676 + T^{2}$$
$79$ $$( -512 + T )^{2}$$
$83$ $$1411344 + T^{2}$$
$89$ $$( -630 + T )^{2}$$
$97$ $$1110916 + T^{2}$$