# Properties

 Label 400.4.a.o Level $400$ Weight $4$ Character orbit 400.a Self dual yes Analytic conductor $23.601$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$23.6007640023$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + 4 q^{3} - 16 q^{7} - 11 q^{9}+O(q^{10})$$ q + 4 * q^3 - 16 * q^7 - 11 * q^9 $$q + 4 q^{3} - 16 q^{7} - 11 q^{9} + 60 q^{11} - 86 q^{13} - 18 q^{17} - 44 q^{19} - 64 q^{21} + 48 q^{23} - 152 q^{27} - 186 q^{29} - 176 q^{31} + 240 q^{33} - 254 q^{37} - 344 q^{39} + 186 q^{41} - 100 q^{43} + 168 q^{47} - 87 q^{49} - 72 q^{51} + 498 q^{53} - 176 q^{57} + 252 q^{59} - 58 q^{61} + 176 q^{63} - 1036 q^{67} + 192 q^{69} - 168 q^{71} - 506 q^{73} - 960 q^{77} - 272 q^{79} - 311 q^{81} + 948 q^{83} - 744 q^{87} - 1014 q^{89} + 1376 q^{91} - 704 q^{93} + 766 q^{97} - 660 q^{99}+O(q^{100})$$ q + 4 * q^3 - 16 * q^7 - 11 * q^9 + 60 * q^11 - 86 * q^13 - 18 * q^17 - 44 * q^19 - 64 * q^21 + 48 * q^23 - 152 * q^27 - 186 * q^29 - 176 * q^31 + 240 * q^33 - 254 * q^37 - 344 * q^39 + 186 * q^41 - 100 * q^43 + 168 * q^47 - 87 * q^49 - 72 * q^51 + 498 * q^53 - 176 * q^57 + 252 * q^59 - 58 * q^61 + 176 * q^63 - 1036 * q^67 + 192 * q^69 - 168 * q^71 - 506 * q^73 - 960 * q^77 - 272 * q^79 - 311 * q^81 + 948 * q^83 - 744 * q^87 - 1014 * q^89 + 1376 * q^91 - 704 * q^93 + 766 * q^97 - 660 * 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 4.00000 0 0 0 −16.0000 0 −11.0000 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 400.4.a.o 1
4.b odd 2 1 100.4.a.a 1
5.b even 2 1 80.4.a.c 1
5.c odd 4 2 400.4.c.j 2
8.b even 2 1 1600.4.a.p 1
8.d odd 2 1 1600.4.a.bl 1
12.b even 2 1 900.4.a.m 1
15.d odd 2 1 720.4.a.k 1
20.d odd 2 1 20.4.a.a 1
20.e even 4 2 100.4.c.a 2
40.e odd 2 1 320.4.a.d 1
40.f even 2 1 320.4.a.k 1
60.h even 2 1 180.4.a.a 1
60.l odd 4 2 900.4.d.k 2
80.k odd 4 2 1280.4.d.n 2
80.q even 4 2 1280.4.d.c 2
140.c even 2 1 980.4.a.c 1
140.p odd 6 2 980.4.i.e 2
140.s even 6 2 980.4.i.n 2
180.n even 6 2 1620.4.i.j 2
180.p odd 6 2 1620.4.i.d 2
220.g even 2 1 2420.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 20.d odd 2 1
80.4.a.c 1 5.b even 2 1
100.4.a.a 1 4.b odd 2 1
100.4.c.a 2 20.e even 4 2
180.4.a.a 1 60.h even 2 1
320.4.a.d 1 40.e odd 2 1
320.4.a.k 1 40.f even 2 1
400.4.a.o 1 1.a even 1 1 trivial
400.4.c.j 2 5.c odd 4 2
720.4.a.k 1 15.d odd 2 1
900.4.a.m 1 12.b even 2 1
900.4.d.k 2 60.l odd 4 2
980.4.a.c 1 140.c even 2 1
980.4.i.e 2 140.p odd 6 2
980.4.i.n 2 140.s even 6 2
1280.4.d.c 2 80.q even 4 2
1280.4.d.n 2 80.k odd 4 2
1600.4.a.p 1 8.b even 2 1
1600.4.a.bl 1 8.d odd 2 1
1620.4.i.d 2 180.p odd 6 2
1620.4.i.j 2 180.n even 6 2
2420.4.a.d 1 220.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(400))$$:

 $$T_{3} - 4$$ T3 - 4 $$T_{7} + 16$$ T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 4$$
$5$ $$T$$
$7$ $$T + 16$$
$11$ $$T - 60$$
$13$ $$T + 86$$
$17$ $$T + 18$$
$19$ $$T + 44$$
$23$ $$T - 48$$
$29$ $$T + 186$$
$31$ $$T + 176$$
$37$ $$T + 254$$
$41$ $$T - 186$$
$43$ $$T + 100$$
$47$ $$T - 168$$
$53$ $$T - 498$$
$59$ $$T - 252$$
$61$ $$T + 58$$
$67$ $$T + 1036$$
$71$ $$T + 168$$
$73$ $$T + 506$$
$79$ $$T + 272$$
$83$ $$T - 948$$
$89$ $$T + 1014$$
$97$ $$T - 766$$