Properties

 Label 400.6.a.f Level $400$ Weight $6$ Character orbit 400.a Self dual yes Analytic conductor $64.154$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$64.1535279252$$ 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 400.6.a.f 1
4.b odd 2 1 200.6.a.c 1
5.b even 2 1 80.6.a.f 1
5.c odd 4 2 400.6.c.h 2
15.d odd 2 1 720.6.a.h 1
20.d odd 2 1 40.6.a.b 1
20.e even 4 2 200.6.c.c 2
40.e odd 2 1 320.6.a.l 1
40.f even 2 1 320.6.a.e 1
60.h even 2 1 360.6.a.b 1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

Hecke characteristic polynomials

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