# Properties

 Label 400.6.a.s Level $400$ Weight $6$ Character orbit 400.a Self dual yes Analytic conductor $64.154$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{31})$$ Defining polynomial: $$x^{2} - 31$$ x^2 - 31 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 20) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{31}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 11 \beta q^{7} - 119 q^{9} +O(q^{10})$$ q + b * q^3 - 11*b * q^7 - 119 * q^9 $$q + \beta q^{3} - 11 \beta q^{7} - 119 q^{9} + 100 q^{11} + 66 \beta q^{13} - 88 \beta q^{17} + 2244 q^{19} - 1364 q^{21} + 307 \beta q^{23} - 362 \beta q^{27} - 7854 q^{29} + 2144 q^{31} + 100 \beta q^{33} - 934 \beta q^{37} + 8184 q^{39} - 7414 q^{41} - 1595 \beta q^{43} - 847 \beta q^{47} - 1803 q^{49} - 10912 q^{51} - 2178 \beta q^{53} + 2244 \beta q^{57} - 25972 q^{59} - 3058 q^{61} + 1309 \beta q^{63} + 5279 \beta q^{67} + 38068 q^{69} - 37608 q^{71} + 2156 \beta q^{73} - 1100 \beta q^{77} - 79728 q^{79} - 15971 q^{81} - 1463 \beta q^{83} - 7854 \beta q^{87} - 826 q^{89} - 90024 q^{91} + 2144 \beta q^{93} + 3376 \beta q^{97} - 11900 q^{99} +O(q^{100})$$ q + b * q^3 - 11*b * q^7 - 119 * q^9 + 100 * q^11 + 66*b * q^13 - 88*b * q^17 + 2244 * q^19 - 1364 * q^21 + 307*b * q^23 - 362*b * q^27 - 7854 * q^29 + 2144 * q^31 + 100*b * q^33 - 934*b * q^37 + 8184 * q^39 - 7414 * q^41 - 1595*b * q^43 - 847*b * q^47 - 1803 * q^49 - 10912 * q^51 - 2178*b * q^53 + 2244*b * q^57 - 25972 * q^59 - 3058 * q^61 + 1309*b * q^63 + 5279*b * q^67 + 38068 * q^69 - 37608 * q^71 + 2156*b * q^73 - 1100*b * q^77 - 79728 * q^79 - 15971 * q^81 - 1463*b * q^83 - 7854*b * q^87 - 826 * q^89 - 90024 * q^91 + 2144*b * q^93 + 3376*b * q^97 - 11900 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 238 q^{9}+O(q^{10})$$ 2 * q - 238 * q^9 $$2 q - 238 q^{9} + 200 q^{11} + 4488 q^{19} - 2728 q^{21} - 15708 q^{29} + 4288 q^{31} + 16368 q^{39} - 14828 q^{41} - 3606 q^{49} - 21824 q^{51} - 51944 q^{59} - 6116 q^{61} + 76136 q^{69} - 75216 q^{71} - 159456 q^{79} - 31942 q^{81} - 1652 q^{89} - 180048 q^{91} - 23800 q^{99}+O(q^{100})$$ 2 * q - 238 * q^9 + 200 * q^11 + 4488 * q^19 - 2728 * q^21 - 15708 * q^29 + 4288 * q^31 + 16368 * q^39 - 14828 * q^41 - 3606 * q^49 - 21824 * q^51 - 51944 * q^59 - 6116 * q^61 + 76136 * q^69 - 75216 * q^71 - 159456 * q^79 - 31942 * q^81 - 1652 * q^89 - 180048 * q^91 - 23800 * 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
 −5.56776 5.56776
0 −11.1355 0 0 0 122.491 0 −119.000 0
1.2 0 11.1355 0 0 0 −122.491 0 −119.000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.a.s 2
4.b odd 2 1 100.6.a.d 2
5.b even 2 1 inner 400.6.a.s 2
5.c odd 4 2 80.6.c.b 2
12.b even 2 1 900.6.a.q 2
15.e even 4 2 720.6.f.d 2
20.d odd 2 1 100.6.a.d 2
20.e even 4 2 20.6.c.a 2
40.i odd 4 2 320.6.c.d 2
40.k even 4 2 320.6.c.e 2
60.h even 2 1 900.6.a.q 2
60.l odd 4 2 180.6.d.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 124$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 124$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 15004$$
$11$ $$(T - 100)^{2}$$
$13$ $$T^{2} - 540144$$
$17$ $$T^{2} - 960256$$
$19$ $$(T - 2244)^{2}$$
$23$ $$T^{2} - 11686876$$
$29$ $$(T + 7854)^{2}$$
$31$ $$(T - 2144)^{2}$$
$37$ $$T^{2} - 108172144$$
$41$ $$(T + 7414)^{2}$$
$43$ $$T^{2} - 315459100$$
$47$ $$T^{2} - 88958716$$
$53$ $$T^{2} - 588216816$$
$59$ $$(T + 25972)^{2}$$
$61$ $$(T + 3058)^{2}$$
$67$ $$T^{2} - 3455612284$$
$71$ $$(T + 37608)^{2}$$
$73$ $$T^{2} - 576393664$$
$79$ $$(T + 79728)^{2}$$
$83$ $$T^{2} - 265405756$$
$89$ $$(T + 826)^{2}$$
$97$ $$T^{2} - 1413274624$$