# Properties

 Label 400.10.a.j Level 400 Weight 10 Character orbit 400.a Self dual yes Analytic conductor 206.014 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$206.014334466$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 174q^{3} + 4658q^{7} + 10593q^{9} + O(q^{10})$$ $$q + 174q^{3} + 4658q^{7} + 10593q^{9} - 28992q^{11} + 164446q^{13} + 594822q^{17} + 295780q^{19} + 810492q^{21} + 2544534q^{23} - 1581660q^{27} - 3722970q^{29} - 2335772q^{31} - 5044608q^{33} - 10840418q^{37} + 28613604q^{39} + 21593862q^{41} + 10832294q^{43} + 5172138q^{47} - 18656643q^{49} + 103499028q^{51} - 98179674q^{53} + 51465720q^{57} - 16162860q^{59} - 43928158q^{61} + 49342194q^{63} - 81557422q^{67} + 442748916q^{69} - 161307732q^{71} + 247147966q^{73} - 135044736q^{77} + 583345720q^{79} - 483710859q^{81} - 14571786q^{83} - 647796780q^{87} + 470133690q^{89} + 765989468q^{91} - 406424328q^{93} + 117838462q^{97} - 307112256q^{99} + O(q^{100})$$

## 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
0 174.000 0 0 0 4658.00 0 10593.0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 400.10.a.j 1
4.b odd 2 1 50.10.a.a 1
5.b even 2 1 80.10.a.a 1
5.c odd 4 2 400.10.c.c 2
20.d odd 2 1 10.10.a.c 1
20.e even 4 2 50.10.b.d 2
40.e odd 2 1 320.10.a.b 1
40.f even 2 1 320.10.a.i 1
60.h even 2 1 90.10.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.10.a.c 1 20.d odd 2 1
50.10.a.a 1 4.b odd 2 1
50.10.b.d 2 20.e even 4 2
80.10.a.a 1 5.b even 2 1
90.10.a.e 1 60.h even 2 1
320.10.a.b 1 40.e odd 2 1
320.10.a.i 1 40.f even 2 1
400.10.a.j 1 1.a even 1 1 trivial
400.10.c.c 2 5.c odd 4 2

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 174$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 174 T + 19683 T^{2}$$
$5$ 
$7$ $$1 - 4658 T + 40353607 T^{2}$$
$11$ $$1 + 28992 T + 2357947691 T^{2}$$
$13$ $$1 - 164446 T + 10604499373 T^{2}$$
$17$ $$1 - 594822 T + 118587876497 T^{2}$$
$19$ $$1 - 295780 T + 322687697779 T^{2}$$
$23$ $$1 - 2544534 T + 1801152661463 T^{2}$$
$29$ $$1 + 3722970 T + 14507145975869 T^{2}$$
$31$ $$1 + 2335772 T + 26439622160671 T^{2}$$
$37$ $$1 + 10840418 T + 129961739795077 T^{2}$$
$41$ $$1 - 21593862 T + 327381934393961 T^{2}$$
$43$ $$1 - 10832294 T + 502592611936843 T^{2}$$
$47$ $$1 - 5172138 T + 1119130473102767 T^{2}$$
$53$ $$1 + 98179674 T + 3299763591802133 T^{2}$$
$59$ $$1 + 16162860 T + 8662995818654939 T^{2}$$
$61$ $$1 + 43928158 T + 11694146092834141 T^{2}$$
$67$ $$1 + 81557422 T + 27206534396294947 T^{2}$$
$71$ $$1 + 161307732 T + 45848500718449031 T^{2}$$
$73$ $$1 - 247147966 T + 58871586708267913 T^{2}$$
$79$ $$1 - 583345720 T + 119851595982618319 T^{2}$$
$83$ $$1 + 14571786 T + 186940255267540403 T^{2}$$
$89$ $$1 - 470133690 T + 350356403707485209 T^{2}$$
$97$ $$1 - 117838462 T + 760231058654565217 T^{2}$$