# Properties

 Label 1638.4.a.d Level $1638$ Weight $4$ Character orbit 1638.a Self dual yes Analytic conductor $96.645$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1638.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$96.6451285894$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} + 4q^{4} - 3q^{5} + 7q^{7} - 8q^{8} + O(q^{10})$$ $$q - 2q^{2} + 4q^{4} - 3q^{5} + 7q^{7} - 8q^{8} + 6q^{10} + 54q^{11} + 13q^{13} - 14q^{14} + 16q^{16} + 96q^{17} - 151q^{19} - 12q^{20} - 108q^{22} - 33q^{23} - 116q^{25} - 26q^{26} + 28q^{28} - 183q^{29} - 331q^{31} - 32q^{32} - 192q^{34} - 21q^{35} - 88q^{37} + 302q^{38} + 24q^{40} + 42q^{41} + 353q^{43} + 216q^{44} + 66q^{46} + 465q^{47} + 49q^{49} + 232q^{50} + 52q^{52} - 195q^{53} - 162q^{55} - 56q^{56} + 366q^{58} - 552q^{59} + 470q^{61} + 662q^{62} + 64q^{64} - 39q^{65} + 254q^{67} + 384q^{68} + 42q^{70} - 132q^{71} - 943q^{73} + 176q^{74} - 604q^{76} + 378q^{77} - 727q^{79} - 48q^{80} - 84q^{82} + 1197q^{83} - 288q^{85} - 706q^{86} - 432q^{88} - 753q^{89} + 91q^{91} - 132q^{92} - 930q^{94} + 453q^{95} + 1037q^{97} - 98q^{98} + O(q^{100})$$

## 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
−2.00000 0 4.00000 −3.00000 0 7.00000 −8.00000 0 6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 1638.4.a.d 1
3.b odd 2 1 182.4.a.b 1
12.b even 2 1 1456.4.a.g 1
21.c even 2 1 1274.4.a.i 1
39.d odd 2 1 2366.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.4.a.b 1 3.b odd 2 1
1274.4.a.i 1 21.c even 2 1
1456.4.a.g 1 12.b even 2 1
1638.4.a.d 1 1.a even 1 1 trivial
2366.4.a.a 1 39.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1638))$$:

 $$T_{5} + 3$$ $$T_{11} - 54$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$3 + T$$
$7$ $$-7 + T$$
$11$ $$-54 + T$$
$13$ $$-13 + T$$
$17$ $$-96 + T$$
$19$ $$151 + T$$
$23$ $$33 + T$$
$29$ $$183 + T$$
$31$ $$331 + T$$
$37$ $$88 + T$$
$41$ $$-42 + T$$
$43$ $$-353 + T$$
$47$ $$-465 + T$$
$53$ $$195 + T$$
$59$ $$552 + T$$
$61$ $$-470 + T$$
$67$ $$-254 + T$$
$71$ $$132 + T$$
$73$ $$943 + T$$
$79$ $$727 + T$$
$83$ $$-1197 + T$$
$89$ $$753 + T$$
$97$ $$-1037 + T$$