# Properties

 Label 1200.4.a.o Level $1200$ Weight $4$ Character orbit 1200.a Self dual yes Analytic conductor $70.802$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.8022920069$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 20 q^{7} + 9 q^{9} + O(q^{10})$$ $$q - 3 q^{3} + 20 q^{7} + 9 q^{9} + 24 q^{11} - 74 q^{13} - 54 q^{17} + 124 q^{19} - 60 q^{21} - 120 q^{23} - 27 q^{27} - 78 q^{29} - 200 q^{31} - 72 q^{33} + 70 q^{37} + 222 q^{39} + 330 q^{41} + 92 q^{43} - 24 q^{47} + 57 q^{49} + 162 q^{51} - 450 q^{53} - 372 q^{57} - 24 q^{59} - 322 q^{61} + 180 q^{63} - 196 q^{67} + 360 q^{69} + 288 q^{71} + 430 q^{73} + 480 q^{77} + 520 q^{79} + 81 q^{81} + 156 q^{83} + 234 q^{87} + 1026 q^{89} - 1480 q^{91} + 600 q^{93} + 286 q^{97} + 216 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −3.00000 0 0 0 20.0000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.a.o 1
4.b odd 2 1 75.4.a.a 1
5.b even 2 1 240.4.a.f 1
5.c odd 4 2 1200.4.f.m 2
12.b even 2 1 225.4.a.g 1
15.d odd 2 1 720.4.a.r 1
20.d odd 2 1 15.4.a.b 1
20.e even 4 2 75.4.b.a 2
40.e odd 2 1 960.4.a.bi 1
40.f even 2 1 960.4.a.l 1
60.h even 2 1 45.4.a.b 1
60.l odd 4 2 225.4.b.d 2
140.c even 2 1 735.4.a.i 1
180.n even 6 2 405.4.e.k 2
180.p odd 6 2 405.4.e.d 2
220.g even 2 1 1815.4.a.a 1
420.o odd 2 1 2205.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 20.d odd 2 1
45.4.a.b 1 60.h even 2 1
75.4.a.a 1 4.b odd 2 1
75.4.b.a 2 20.e even 4 2
225.4.a.g 1 12.b even 2 1
225.4.b.d 2 60.l odd 4 2
240.4.a.f 1 5.b even 2 1
405.4.e.d 2 180.p odd 6 2
405.4.e.k 2 180.n even 6 2
720.4.a.r 1 15.d odd 2 1
735.4.a.i 1 140.c even 2 1
960.4.a.l 1 40.f even 2 1
960.4.a.bi 1 40.e odd 2 1
1200.4.a.o 1 1.a even 1 1 trivial
1200.4.f.m 2 5.c odd 4 2
1815.4.a.a 1 220.g even 2 1
2205.4.a.c 1 420.o odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1200))$$:

 $$T_{7} - 20$$ $$T_{11} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$T$$
$7$ $$-20 + T$$
$11$ $$-24 + T$$
$13$ $$74 + T$$
$17$ $$54 + T$$
$19$ $$-124 + T$$
$23$ $$120 + T$$
$29$ $$78 + T$$
$31$ $$200 + T$$
$37$ $$-70 + T$$
$41$ $$-330 + T$$
$43$ $$-92 + T$$
$47$ $$24 + T$$
$53$ $$450 + T$$
$59$ $$24 + T$$
$61$ $$322 + T$$
$67$ $$196 + T$$
$71$ $$-288 + T$$
$73$ $$-430 + T$$
$79$ $$-520 + T$$
$83$ $$-156 + T$$
$89$ $$-1026 + T$$
$97$ $$-286 + T$$
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