# Properties

 Label 200.6.a.d Level $200$ Weight $6$ Character orbit 200.a Self dual yes Analytic conductor $32.077$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0767639626$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 18 q^{3} - 242 q^{7} + 81 q^{9}+O(q^{10})$$ q + 18 * q^3 - 242 * q^7 + 81 * q^9 $$q + 18 q^{3} - 242 q^{7} + 81 q^{9} + 656 q^{11} + 206 q^{13} - 1690 q^{17} - 1364 q^{19} - 4356 q^{21} - 2198 q^{23} - 2916 q^{27} - 2218 q^{29} - 1700 q^{31} + 11808 q^{33} + 846 q^{37} + 3708 q^{39} - 1818 q^{41} - 10534 q^{43} - 12074 q^{47} + 41757 q^{49} - 30420 q^{51} - 32586 q^{53} - 24552 q^{57} + 8668 q^{59} - 34670 q^{61} - 19602 q^{63} + 47566 q^{67} - 39564 q^{69} + 948 q^{71} + 63102 q^{73} - 158752 q^{77} + 46536 q^{79} - 72171 q^{81} + 88778 q^{83} - 39924 q^{87} - 104934 q^{89} - 49852 q^{91} - 30600 q^{93} + 36254 q^{97} + 53136 q^{99}+O(q^{100})$$ q + 18 * q^3 - 242 * q^7 + 81 * q^9 + 656 * q^11 + 206 * q^13 - 1690 * q^17 - 1364 * q^19 - 4356 * q^21 - 2198 * q^23 - 2916 * q^27 - 2218 * q^29 - 1700 * q^31 + 11808 * q^33 + 846 * q^37 + 3708 * q^39 - 1818 * q^41 - 10534 * q^43 - 12074 * q^47 + 41757 * q^49 - 30420 * q^51 - 32586 * q^53 - 24552 * q^57 + 8668 * q^59 - 34670 * q^61 - 19602 * q^63 + 47566 * q^67 - 39564 * q^69 + 948 * q^71 + 63102 * q^73 - 158752 * q^77 + 46536 * q^79 - 72171 * q^81 + 88778 * q^83 - 39924 * q^87 - 104934 * q^89 - 49852 * q^91 - 30600 * q^93 + 36254 * q^97 + 53136 * q^99

## 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 18.0000 0 0 0 −242.000 0 81.0000 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
$$2$$ $$1$$
$$5$$ $$1$$

## 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 200.6.a.d 1
4.b odd 2 1 400.6.a.b 1
5.b even 2 1 40.6.a.a 1
5.c odd 4 2 200.6.c.b 2
15.d odd 2 1 360.6.a.i 1
20.d odd 2 1 80.6.a.g 1
20.e even 4 2 400.6.c.e 2
40.e odd 2 1 320.6.a.d 1
40.f even 2 1 320.6.a.m 1
60.h even 2 1 720.6.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.a 1 5.b even 2 1
80.6.a.g 1 20.d odd 2 1
200.6.a.d 1 1.a even 1 1 trivial
200.6.c.b 2 5.c odd 4 2
320.6.a.d 1 40.e odd 2 1
320.6.a.m 1 40.f even 2 1
360.6.a.i 1 15.d odd 2 1
400.6.a.b 1 4.b odd 2 1
400.6.c.e 2 20.e even 4 2
720.6.a.k 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 18$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(200))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 18$$
$5$ $$T$$
$7$ $$T + 242$$
$11$ $$T - 656$$
$13$ $$T - 206$$
$17$ $$T + 1690$$
$19$ $$T + 1364$$
$23$ $$T + 2198$$
$29$ $$T + 2218$$
$31$ $$T + 1700$$
$37$ $$T - 846$$
$41$ $$T + 1818$$
$43$ $$T + 10534$$
$47$ $$T + 12074$$
$53$ $$T + 32586$$
$59$ $$T - 8668$$
$61$ $$T + 34670$$
$67$ $$T - 47566$$
$71$ $$T - 948$$
$73$ $$T - 63102$$
$79$ $$T - 46536$$
$83$ $$T - 88778$$
$89$ $$T + 104934$$
$97$ $$T - 36254$$
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