# Properties

 Label 38.6.a.a Level $38$ Weight $6$ Character orbit 38.a Self dual yes Analytic conductor $6.095$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 38.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.09458515289$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{2} - 6q^{3} + 16q^{4} + 31q^{5} + 24q^{6} - 27q^{7} - 64q^{8} - 207q^{9} + O(q^{10})$$ $$q - 4q^{2} - 6q^{3} + 16q^{4} + 31q^{5} + 24q^{6} - 27q^{7} - 64q^{8} - 207q^{9} - 124q^{10} - 323q^{11} - 96q^{12} - 676q^{13} + 108q^{14} - 186q^{15} + 256q^{16} - 1107q^{17} + 828q^{18} - 361q^{19} + 496q^{20} + 162q^{21} + 1292q^{22} + 1384q^{23} + 384q^{24} - 2164q^{25} + 2704q^{26} + 2700q^{27} - 432q^{28} + 2870q^{29} + 744q^{30} + 1372q^{31} - 1024q^{32} + 1938q^{33} + 4428q^{34} - 837q^{35} - 3312q^{36} - 7982q^{37} + 1444q^{38} + 4056q^{39} - 1984q^{40} + 1202q^{41} - 648q^{42} - 9911q^{43} - 5168q^{44} - 6417q^{45} - 5536q^{46} + 3463q^{47} - 1536q^{48} - 16078q^{49} + 8656q^{50} + 6642q^{51} - 10816q^{52} + 17764q^{53} - 10800q^{54} - 10013q^{55} + 1728q^{56} + 2166q^{57} - 11480q^{58} + 27270q^{59} - 2976q^{60} + 20867q^{61} - 5488q^{62} + 5589q^{63} + 4096q^{64} - 20956q^{65} - 7752q^{66} + 15228q^{67} - 17712q^{68} - 8304q^{69} + 3348q^{70} + 40642q^{71} + 13248q^{72} - 66711q^{73} + 31928q^{74} + 12984q^{75} - 5776q^{76} + 8721q^{77} - 16224q^{78} + 68960q^{79} + 7936q^{80} + 34101q^{81} - 4808q^{82} - 12396q^{83} + 2592q^{84} - 34317q^{85} + 39644q^{86} - 17220q^{87} + 20672q^{88} + 41220q^{89} + 25668q^{90} + 18252q^{91} + 22144q^{92} - 8232q^{93} - 13852q^{94} - 11191q^{95} + 6144q^{96} - 113432q^{97} + 64312q^{98} + 66861q^{99} + O(q^{100})$$

## 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
−4.00000 −6.00000 16.0000 31.0000 24.0000 −27.0000 −64.0000 −207.000 −124.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$19$$ $$1$$

## 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 38.6.a.a 1
3.b odd 2 1 342.6.a.e 1
4.b odd 2 1 304.6.a.c 1
5.b even 2 1 950.6.a.b 1
19.b odd 2 1 722.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.a.a 1 1.a even 1 1 trivial
304.6.a.c 1 4.b odd 2 1
342.6.a.e 1 3.b odd 2 1
722.6.a.b 1 19.b odd 2 1
950.6.a.b 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 6$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(38))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$6 + T$$
$5$ $$-31 + T$$
$7$ $$27 + T$$
$11$ $$323 + T$$
$13$ $$676 + T$$
$17$ $$1107 + T$$
$19$ $$361 + T$$
$23$ $$-1384 + T$$
$29$ $$-2870 + T$$
$31$ $$-1372 + T$$
$37$ $$7982 + T$$
$41$ $$-1202 + T$$
$43$ $$9911 + T$$
$47$ $$-3463 + T$$
$53$ $$-17764 + T$$
$59$ $$-27270 + T$$
$61$ $$-20867 + T$$
$67$ $$-15228 + T$$
$71$ $$-40642 + T$$
$73$ $$66711 + T$$
$79$ $$-68960 + T$$
$83$ $$12396 + T$$
$89$ $$-41220 + T$$
$97$ $$113432 + T$$
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