# Properties

 Label 400.6.a.x Level $400$ Weight $6$ Character orbit 400.a Self dual yes Analytic conductor $64.154$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.47217.1 Defining polynomial: $$x^{3} - x^{2} - 38x - 24$$ x^3 - x^2 - 38*x - 24 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 5$$ Twist minimal: no (minimal twist has level 200) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 + 24) q^{7} + ( - \beta_{2} + 5 \beta_1 + 50) q^{9}+O(q^{10})$$ q - b1 * q^3 + (-b2 - b1 + 24) * q^7 + (-b2 + 5*b1 + 50) * q^9 $$q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 + 24) q^{7} + ( - \beta_{2} + 5 \beta_1 + 50) q^{9} + ( - \beta_{2} + 8 \beta_1 + 4) q^{11} + (\beta_{2} + 31 \beta_1 - 76) q^{13} + ( - 5 \beta_{2} + 49 \beta_1 + 393) q^{17} + ( - 6 \beta_{2} - 87 \beta_1 - 376) q^{19} + ( - 5 \beta_{2} - 147 \beta_1 + 370) q^{21} + (6 \beta_{2} - 204 \beta_1 + 896) q^{23} + (\beta_{2} + 40 \beta_1 - 1388) q^{27} + ( - 5 \beta_{2} + 13 \beta_1 + 3968) q^{29} + (2 \beta_{2} - 220 \beta_1 - 2408) q^{31} + (4 \beta_{2} - 172 \beta_1 - 2267) q^{33} + (39 \beta_{2} + 201 \beta_1 + 4842) q^{37} + (35 \beta_{2} + 49 \beta_1 - 9160) q^{39} + (44 \beta_{2} - 292 \beta_1 + 1157) q^{41} + ( - 41 \beta_{2} + 241 \beta_1 + 13620) q^{43} + (31 \beta_{2} + 733 \beta_1 - 9984) q^{47} + (72 \beta_{2} - 888 \beta_1 + 22413) q^{49} + (29 \beta_{2} - 1278 \beta_1 - 13972) q^{51} + ( - 76 \beta_{2} - 1468 \beta_1 - 3778) q^{53} + ( - 111 \beta_{2} + 43 \beta_1 + 25953) q^{57} + ( - 35 \beta_{2} + 151 \beta_1 - 21676) q^{59} + ( - 155 \beta_{2} + 19 \beta_1 + 7538) q^{61} + (76 \beta_{2} - 32 \beta_1 + 37624) q^{63} + ( - 105 \beta_{2} + 162 \beta_1 - 8852) q^{67} + ( - 180 \beta_{2} + 892 \beta_1 + 59310) q^{69} + ( - 135 \beta_{2} + 1299 \beta_1 - 29280) q^{71} + (257 \beta_{2} + 2363 \beta_1 + 1895) q^{73} + (137 \beta_{2} + 455 \beta_1 + 35410) q^{77} + ( - 113 \beta_{2} - 1181 \beta_1 - 6896) q^{79} + (287 \beta_{2} + 101 \beta_1 - 23947) q^{81} + (43 \beta_{2} - 1310 \beta_1 + 71540) q^{83} + ( - 7 \beta_{2} - 4673 \beta_1 - 3424) q^{87} + ( - 41 \beta_{2} + 1141 \beta_1 + 60427) q^{89} + (130 \beta_{2} + 5350 \beta_1 - 51568) q^{91} + ( - 212 \beta_{2} + 3764 \beta_1 + 64306) q^{93} + ( - 82 \beta_{2} - 4846 \beta_1 + 27090) q^{97} + (87 \beta_{2} + 1695 \beta_1 + 49116) q^{99}+O(q^{100})$$ q - b1 * q^3 + (-b2 - b1 + 24) * q^7 + (-b2 + 5*b1 + 50) * q^9 + (-b2 + 8*b1 + 4) * q^11 + (b2 + 31*b1 - 76) * q^13 + (-5*b2 + 49*b1 + 393) * q^17 + (-6*b2 - 87*b1 - 376) * q^19 + (-5*b2 - 147*b1 + 370) * q^21 + (6*b2 - 204*b1 + 896) * q^23 + (b2 + 40*b1 - 1388) * q^27 + (-5*b2 + 13*b1 + 3968) * q^29 + (2*b2 - 220*b1 - 2408) * q^31 + (4*b2 - 172*b1 - 2267) * q^33 + (39*b2 + 201*b1 + 4842) * q^37 + (35*b2 + 49*b1 - 9160) * q^39 + (44*b2 - 292*b1 + 1157) * q^41 + (-41*b2 + 241*b1 + 13620) * q^43 + (31*b2 + 733*b1 - 9984) * q^47 + (72*b2 - 888*b1 + 22413) * q^49 + (29*b2 - 1278*b1 - 13972) * q^51 + (-76*b2 - 1468*b1 - 3778) * q^53 + (-111*b2 + 43*b1 + 25953) * q^57 + (-35*b2 + 151*b1 - 21676) * q^59 + (-155*b2 + 19*b1 + 7538) * q^61 + (76*b2 - 32*b1 + 37624) * q^63 + (-105*b2 + 162*b1 - 8852) * q^67 + (-180*b2 + 892*b1 + 59310) * q^69 + (-135*b2 + 1299*b1 - 29280) * q^71 + (257*b2 + 2363*b1 + 1895) * q^73 + (137*b2 + 455*b1 + 35410) * q^77 + (-113*b2 - 1181*b1 - 6896) * q^79 + (287*b2 + 101*b1 - 23947) * q^81 + (43*b2 - 1310*b1 + 71540) * q^83 + (-7*b2 - 4673*b1 - 3424) * q^87 + (-41*b2 + 1141*b1 + 60427) * q^89 + (130*b2 + 5350*b1 - 51568) * q^91 + (-212*b2 + 3764*b1 + 64306) * q^93 + (-82*b2 - 4846*b1 + 27090) * q^97 + (87*b2 + 1695*b1 + 49116) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} + 70 q^{7} + 154 q^{9}+O(q^{10})$$ 3 * q - q^3 + 70 * q^7 + 154 * q^9 $$3 q - q^{3} + 70 q^{7} + 154 q^{9} + 19 q^{11} - 196 q^{13} + 1223 q^{17} - 1221 q^{19} + 958 q^{21} + 2490 q^{23} - 4123 q^{27} + 11912 q^{29} - 7442 q^{31} - 6969 q^{33} + 14766 q^{37} - 27396 q^{39} + 3223 q^{41} + 41060 q^{43} - 29188 q^{47} + 66423 q^{49} - 43165 q^{51} - 12878 q^{53} + 77791 q^{57} - 64912 q^{59} + 22478 q^{61} + 112916 q^{63} - 26499 q^{67} + 178642 q^{69} - 86676 q^{71} + 8305 q^{73} + 106822 q^{77} - 21982 q^{79} - 71453 q^{81} + 213353 q^{83} - 14952 q^{87} + 182381 q^{89} - 149224 q^{91} + 196470 q^{93} + 76342 q^{97} + 149130 q^{99}+O(q^{100})$$ 3 * q - q^3 + 70 * q^7 + 154 * q^9 + 19 * q^11 - 196 * q^13 + 1223 * q^17 - 1221 * q^19 + 958 * q^21 + 2490 * q^23 - 4123 * q^27 + 11912 * q^29 - 7442 * q^31 - 6969 * q^33 + 14766 * q^37 - 27396 * q^39 + 3223 * q^41 + 41060 * q^43 - 29188 * q^47 + 66423 * q^49 - 43165 * q^51 - 12878 * q^53 + 77791 * q^57 - 64912 * q^59 + 22478 * q^61 + 112916 * q^63 - 26499 * q^67 + 178642 * q^69 - 86676 * q^71 + 8305 * q^73 + 106822 * q^77 - 21982 * q^79 - 71453 * q^81 + 213353 * q^83 - 14952 * q^87 + 182381 * q^89 - 149224 * q^91 + 196470 * q^93 + 76342 * q^97 + 149130 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 38x - 24$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{2} + 3\nu + 25$$ -v^2 + 3*v + 25 $$\beta_{2}$$ $$=$$ $$3\nu^{2} + 31\nu - 87$$ 3*v^2 + 31*v - 87
 $$\nu$$ $$=$$ $$( \beta_{2} + 3\beta _1 + 12 ) / 40$$ (b2 + 3*b1 + 12) / 40 $$\nu^{2}$$ $$=$$ $$( 3\beta_{2} - 31\beta _1 + 1036 ) / 40$$ (3*b2 - 31*b1 + 1036) / 40

## 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
 −0.649919 6.95752 −5.30760
0 −22.6278 0 0 0 107.252 0 269.020 0
1.2 0 2.53448 0 0 0 −247.370 0 −236.576 0
1.3 0 19.0934 0 0 0 210.117 0 121.557 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.x 3
4.b odd 2 1 200.6.a.i yes 3
5.b even 2 1 400.6.a.y 3
5.c odd 4 2 400.6.c.p 6
20.d odd 2 1 200.6.a.h 3
20.e even 4 2 200.6.c.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.a.h 3 20.d odd 2 1
200.6.a.i yes 3 4.b odd 2 1
200.6.c.g 6 20.e even 4 2
400.6.a.x 3 1.a even 1 1 trivial
400.6.a.y 3 5.b even 2 1
400.6.c.p 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + T_{3}^{2} - 441T_{3} + 1095$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 441 T + 1095$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 70 T^{2} - 55972 T + 5574616$$
$11$ $$T^{3} - 19 T^{2} - 84401 T + 1542819$$
$13$ $$T^{3} + 196 T^{2} + \cdots + 51768768$$
$17$ $$T^{3} - 1223 T^{2} + \cdots + 654046475$$
$19$ $$T^{3} + 1221 T^{2} + \cdots - 7033288181$$
$23$ $$T^{3} - 2490 T^{2} + \cdots + 50437840744$$
$29$ $$T^{3} - 11912 T^{2} + \cdots - 55993024512$$
$31$ $$T^{3} + 7442 T^{2} + \cdots - 14434811400$$
$37$ $$T^{3} - 14766 T^{2} + \cdots + 435216240216$$
$41$ $$T^{3} - 3223 T^{2} + \cdots - 86972031621$$
$43$ $$T^{3} - 41060 T^{2} + \cdots - 660952050112$$
$47$ $$T^{3} + 29188 T^{2} + \cdots - 324350860864$$
$53$ $$T^{3} + 12878 T^{2} + \cdots - 22325036001624$$
$59$ $$T^{3} + 64912 T^{2} + \cdots + 8618337690624$$
$61$ $$T^{3} - 22478 T^{2} + \cdots + 28184724893400$$
$67$ $$T^{3} + 26499 T^{2} + \cdots + 1261173627437$$
$71$ $$T^{3} + 86676 T^{2} + \cdots - 31597978616640$$
$73$ $$T^{3} + \cdots + 160799137840701$$
$79$ $$T^{3} + 21982 T^{2} + \cdots - 26059657326840$$
$83$ $$T^{3} + \cdots - 289154781250911$$
$89$ $$T^{3} + \cdots - 190810675088559$$
$97$ $$T^{3} + \cdots + 166573103686456$$