# Properties

 Label 400.6.a.m Level $400$ Weight $6$ Character orbit 400.a Self dual yes Analytic conductor $64.154$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 22 q^{3} + 218 q^{7} + 241 q^{9}+O(q^{10})$$ q + 22 * q^3 + 218 * q^7 + 241 * q^9 $$q + 22 q^{3} + 218 q^{7} + 241 q^{9} + 480 q^{11} + 622 q^{13} - 186 q^{17} + 1204 q^{19} + 4796 q^{21} - 3186 q^{23} - 44 q^{27} + 5526 q^{29} - 9356 q^{31} + 10560 q^{33} - 5618 q^{37} + 13684 q^{39} - 14394 q^{41} - 370 q^{43} + 16146 q^{47} + 30717 q^{49} - 4092 q^{51} + 4374 q^{53} + 26488 q^{57} + 11748 q^{59} + 13202 q^{61} + 52538 q^{63} - 11542 q^{67} - 70092 q^{69} + 29532 q^{71} - 33698 q^{73} + 104640 q^{77} - 31208 q^{79} - 59531 q^{81} - 38466 q^{83} + 121572 q^{87} + 119514 q^{89} + 135596 q^{91} - 205832 q^{93} - 94658 q^{97} + 115680 q^{99}+O(q^{100})$$ q + 22 * q^3 + 218 * q^7 + 241 * q^9 + 480 * q^11 + 622 * q^13 - 186 * q^17 + 1204 * q^19 + 4796 * q^21 - 3186 * q^23 - 44 * q^27 + 5526 * q^29 - 9356 * q^31 + 10560 * q^33 - 5618 * q^37 + 13684 * q^39 - 14394 * q^41 - 370 * q^43 + 16146 * q^47 + 30717 * q^49 - 4092 * q^51 + 4374 * q^53 + 26488 * q^57 + 11748 * q^59 + 13202 * q^61 + 52538 * q^63 - 11542 * q^67 - 70092 * q^69 + 29532 * q^71 - 33698 * q^73 + 104640 * q^77 - 31208 * q^79 - 59531 * q^81 - 38466 * q^83 + 121572 * q^87 + 119514 * q^89 + 135596 * q^91 - 205832 * q^93 - 94658 * q^97 + 115680 * 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 22.0000 0 0 0 218.000 0 241.000 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.m 1
4.b odd 2 1 100.6.a.a 1
5.b even 2 1 80.6.a.b 1
5.c odd 4 2 400.6.c.c 2
12.b even 2 1 900.6.a.b 1
15.d odd 2 1 720.6.a.l 1
20.d odd 2 1 20.6.a.a 1
20.e even 4 2 100.6.c.a 2
40.e odd 2 1 320.6.a.c 1
40.f even 2 1 320.6.a.n 1
60.h even 2 1 180.6.a.e 1
60.l odd 4 2 900.6.d.h 2
140.c even 2 1 980.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 20.d odd 2 1
80.6.a.b 1 5.b even 2 1
100.6.a.a 1 4.b odd 2 1
100.6.c.a 2 20.e even 4 2
180.6.a.e 1 60.h even 2 1
320.6.a.c 1 40.e odd 2 1
320.6.a.n 1 40.f even 2 1
400.6.a.m 1 1.a even 1 1 trivial
400.6.c.c 2 5.c odd 4 2
720.6.a.l 1 15.d odd 2 1
900.6.a.b 1 12.b even 2 1
900.6.d.h 2 60.l odd 4 2
980.6.a.b 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 22$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 22$$
$5$ $$T$$
$7$ $$T - 218$$
$11$ $$T - 480$$
$13$ $$T - 622$$
$17$ $$T + 186$$
$19$ $$T - 1204$$
$23$ $$T + 3186$$
$29$ $$T - 5526$$
$31$ $$T + 9356$$
$37$ $$T + 5618$$
$41$ $$T + 14394$$
$43$ $$T + 370$$
$47$ $$T - 16146$$
$53$ $$T - 4374$$
$59$ $$T - 11748$$
$61$ $$T - 13202$$
$67$ $$T + 11542$$
$71$ $$T - 29532$$
$73$ $$T + 33698$$
$79$ $$T + 31208$$
$83$ $$T + 38466$$
$89$ $$T - 119514$$
$97$ $$T + 94658$$