# Properties

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

# Related objects

## 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

 $$f(q)$$ $$=$$ $$q + 8 q^{3} + 108 q^{7} - 179 q^{9}+O(q^{10})$$ q + 8 * q^3 + 108 * q^7 - 179 * q^9 $$q + 8 q^{3} + 108 q^{7} - 179 q^{9} - 604 q^{11} + 306 q^{13} - 930 q^{17} - 1324 q^{19} + 864 q^{21} + 852 q^{23} - 3376 q^{27} + 5902 q^{29} - 3320 q^{31} - 4832 q^{33} - 10774 q^{37} + 2448 q^{39} - 17958 q^{41} - 9264 q^{43} + 9796 q^{47} - 5143 q^{49} - 7440 q^{51} + 31434 q^{53} - 10592 q^{57} + 33228 q^{59} - 40210 q^{61} - 19332 q^{63} - 58864 q^{67} + 6816 q^{69} - 55312 q^{71} - 27258 q^{73} - 65232 q^{77} + 31456 q^{79} + 16489 q^{81} - 24552 q^{83} + 47216 q^{87} - 90854 q^{89} + 33048 q^{91} - 26560 q^{93} - 154706 q^{97} + 108116 q^{99}+O(q^{100})$$ q + 8 * q^3 + 108 * q^7 - 179 * q^9 - 604 * q^11 + 306 * q^13 - 930 * q^17 - 1324 * q^19 + 864 * q^21 + 852 * q^23 - 3376 * q^27 + 5902 * q^29 - 3320 * q^31 - 4832 * q^33 - 10774 * q^37 + 2448 * q^39 - 17958 * q^41 - 9264 * q^43 + 9796 * q^47 - 5143 * q^49 - 7440 * q^51 + 31434 * q^53 - 10592 * q^57 + 33228 * q^59 - 40210 * q^61 - 19332 * q^63 - 58864 * q^67 + 6816 * q^69 - 55312 * q^71 - 27258 * q^73 - 65232 * q^77 + 31456 * q^79 + 16489 * q^81 - 24552 * q^83 + 47216 * q^87 - 90854 * q^89 + 33048 * q^91 - 26560 * q^93 - 154706 * q^97 + 108116 * 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 8.00000 0 0 0 108.000 0 −179.000 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.c 1
4.b odd 2 1 400.6.a.f 1
5.b even 2 1 40.6.a.b 1
5.c odd 4 2 200.6.c.c 2
15.d odd 2 1 360.6.a.b 1
20.d odd 2 1 80.6.a.f 1
20.e even 4 2 400.6.c.h 2
40.e odd 2 1 320.6.a.e 1
40.f even 2 1 320.6.a.l 1
60.h even 2 1 720.6.a.h 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(200))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 8$$
$5$ $$T$$
$7$ $$T - 108$$
$11$ $$T + 604$$
$13$ $$T - 306$$
$17$ $$T + 930$$
$19$ $$T + 1324$$
$23$ $$T - 852$$
$29$ $$T - 5902$$
$31$ $$T + 3320$$
$37$ $$T + 10774$$
$41$ $$T + 17958$$
$43$ $$T + 9264$$
$47$ $$T - 9796$$
$53$ $$T - 31434$$
$59$ $$T - 33228$$
$61$ $$T + 40210$$
$67$ $$T + 58864$$
$71$ $$T + 55312$$
$73$ $$T + 27258$$
$79$ $$T - 31456$$
$83$ $$T + 24552$$
$89$ $$T + 90854$$
$97$ $$T + 154706$$