# Properties

 Label 400.6.a.n Level $400$ Weight $6$ Character orbit 400.a Self dual yes Analytic conductor $64.154$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more about

## 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 10) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 24q^{3} - 172q^{7} + 333q^{9} + O(q^{10})$$ $$q + 24q^{3} - 172q^{7} + 333q^{9} - 132q^{11} + 946q^{13} + 222q^{17} - 500q^{19} - 4128q^{21} + 3564q^{23} + 2160q^{27} + 2190q^{29} - 2312q^{31} - 3168q^{33} + 11242q^{37} + 22704q^{39} + 1242q^{41} + 20624q^{43} + 6588q^{47} + 12777q^{49} + 5328q^{51} + 21066q^{53} - 12000q^{57} - 7980q^{59} + 16622q^{61} - 57276q^{63} + 1808q^{67} + 85536q^{69} + 24528q^{71} - 20474q^{73} + 22704q^{77} + 46240q^{79} - 29079q^{81} - 51576q^{83} + 52560q^{87} - 110310q^{89} - 162712q^{91} - 55488q^{93} + 78382q^{97} - 43956q^{99} + O(q^{100})$$

## 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 24.0000 0 0 0 −172.000 0 333.000 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.n 1
4.b odd 2 1 50.6.a.d 1
5.b even 2 1 80.6.a.a 1
5.c odd 4 2 400.6.c.b 2
12.b even 2 1 450.6.a.l 1
15.d odd 2 1 720.6.a.j 1
20.d odd 2 1 10.6.a.b 1
20.e even 4 2 50.6.b.a 2
40.e odd 2 1 320.6.a.b 1
40.f even 2 1 320.6.a.o 1
60.h even 2 1 90.6.a.d 1
60.l odd 4 2 450.6.c.h 2
140.c even 2 1 490.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 20.d odd 2 1
50.6.a.d 1 4.b odd 2 1
50.6.b.a 2 20.e even 4 2
80.6.a.a 1 5.b even 2 1
90.6.a.d 1 60.h even 2 1
320.6.a.b 1 40.e odd 2 1
320.6.a.o 1 40.f even 2 1
400.6.a.n 1 1.a even 1 1 trivial
400.6.c.b 2 5.c odd 4 2
450.6.a.l 1 12.b even 2 1
450.6.c.h 2 60.l odd 4 2
490.6.a.a 1 140.c even 2 1
720.6.a.j 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 24$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.