# Properties

 Label 400.4.a.s Level $400$ Weight $4$ Character orbit 400.a Self dual yes Analytic conductor $23.601$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 7 q^{3} + 6 q^{7} + 22 q^{9} + O(q^{10})$$ $$q + 7 q^{3} + 6 q^{7} + 22 q^{9} + 43 q^{11} + 28 q^{13} - 91 q^{17} + 35 q^{19} + 42 q^{21} + 162 q^{23} - 35 q^{27} + 160 q^{29} - 42 q^{31} + 301 q^{33} + 314 q^{37} + 196 q^{39} - 203 q^{41} + 92 q^{43} + 196 q^{47} - 307 q^{49} - 637 q^{51} - 82 q^{53} + 245 q^{57} + 280 q^{59} - 518 q^{61} + 132 q^{63} + 141 q^{67} + 1134 q^{69} - 412 q^{71} + 763 q^{73} + 258 q^{77} - 510 q^{79} - 839 q^{81} + 777 q^{83} + 1120 q^{87} - 945 q^{89} + 168 q^{91} - 294 q^{93} - 1246 q^{97} + 946 q^{99} + O(q^{100})$$

## 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 7.00000 0 0 0 6.00000 0 22.0000 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.s 1
4.b odd 2 1 25.4.a.a 1
5.b even 2 1 400.4.a.c 1
5.c odd 4 2 400.4.c.e 2
8.b even 2 1 1600.4.a.h 1
8.d odd 2 1 1600.4.a.bt 1
12.b even 2 1 225.4.a.e 1
20.d odd 2 1 25.4.a.b yes 1
20.e even 4 2 25.4.b.b 2
28.d even 2 1 1225.4.a.h 1
40.e odd 2 1 1600.4.a.i 1
40.f even 2 1 1600.4.a.bs 1
60.h even 2 1 225.4.a.c 1
60.l odd 4 2 225.4.b.f 2
140.c even 2 1 1225.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 4.b odd 2 1
25.4.a.b yes 1 20.d odd 2 1
25.4.b.b 2 20.e even 4 2
225.4.a.c 1 60.h even 2 1
225.4.a.e 1 12.b even 2 1
225.4.b.f 2 60.l odd 4 2
400.4.a.c 1 5.b even 2 1
400.4.a.s 1 1.a even 1 1 trivial
400.4.c.e 2 5.c odd 4 2
1225.4.a.h 1 28.d even 2 1
1225.4.a.i 1 140.c even 2 1
1600.4.a.h 1 8.b even 2 1
1600.4.a.i 1 40.e odd 2 1
1600.4.a.bs 1 40.f even 2 1
1600.4.a.bt 1 8.d odd 2 1

## Hecke kernels

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

 $$T_{3} - 7$$ $$T_{7} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-7 + T$$
$5$ $$T$$
$7$ $$-6 + T$$
$11$ $$-43 + T$$
$13$ $$-28 + T$$
$17$ $$91 + T$$
$19$ $$-35 + T$$
$23$ $$-162 + T$$
$29$ $$-160 + T$$
$31$ $$42 + T$$
$37$ $$-314 + T$$
$41$ $$203 + T$$
$43$ $$-92 + T$$
$47$ $$-196 + T$$
$53$ $$82 + T$$
$59$ $$-280 + T$$
$61$ $$518 + T$$
$67$ $$-141 + T$$
$71$ $$412 + T$$
$73$ $$-763 + T$$
$79$ $$510 + T$$
$83$ $$-777 + T$$
$89$ $$945 + T$$
$97$ $$1246 + T$$