# Properties

 Label 1638.4.a.a Level $1638$ Weight $4$ Character orbit 1638.a Self dual yes Analytic conductor $96.645$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} - 16 q^{5} + 7 q^{7} - 8 q^{8} + O(q^{10})$$ $$q - 2 q^{2} + 4 q^{4} - 16 q^{5} + 7 q^{7} - 8 q^{8} + 32 q^{10} + 15 q^{11} - 13 q^{13} - 14 q^{14} + 16 q^{16} + 44 q^{17} - 138 q^{19} - 64 q^{20} - 30 q^{22} - 111 q^{23} + 131 q^{25} + 26 q^{26} + 28 q^{28} + 12 q^{29} + 215 q^{31} - 32 q^{32} - 88 q^{34} - 112 q^{35} + 55 q^{37} + 276 q^{38} + 128 q^{40} + 133 q^{41} - 180 q^{43} + 60 q^{44} + 222 q^{46} - 471 q^{47} + 49 q^{49} - 262 q^{50} - 52 q^{52} + 260 q^{53} - 240 q^{55} - 56 q^{56} - 24 q^{58} - 110 q^{59} - 271 q^{61} - 430 q^{62} + 64 q^{64} + 208 q^{65} - 799 q^{67} + 176 q^{68} + 224 q^{70} - 912 q^{71} + 747 q^{73} - 110 q^{74} - 552 q^{76} + 105 q^{77} - 883 q^{79} - 256 q^{80} - 266 q^{82} + 924 q^{83} - 704 q^{85} + 360 q^{86} - 120 q^{88} - 142 q^{89} - 91 q^{91} - 444 q^{92} + 942 q^{94} + 2208 q^{95} - 1407 q^{97} - 98 q^{98} + O(q^{100})$$

## 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 −16.0000 0 7.00000 −8.00000 0 32.0000
 $$n$$: 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.a 1
3.b odd 2 1 182.4.a.d 1
12.b even 2 1 1456.4.a.c 1
21.c even 2 1 1274.4.a.c 1
39.d odd 2 1 2366.4.a.e 1

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

 $$T_{5} + 16$$ $$T_{11} - 15$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$16 + T$$
$7$ $$-7 + T$$
$11$ $$-15 + T$$
$13$ $$13 + T$$
$17$ $$-44 + T$$
$19$ $$138 + T$$
$23$ $$111 + T$$
$29$ $$-12 + T$$
$31$ $$-215 + T$$
$37$ $$-55 + T$$
$41$ $$-133 + T$$
$43$ $$180 + T$$
$47$ $$471 + T$$
$53$ $$-260 + T$$
$59$ $$110 + T$$
$61$ $$271 + T$$
$67$ $$799 + T$$
$71$ $$912 + T$$
$73$ $$-747 + T$$
$79$ $$883 + T$$
$83$ $$-924 + T$$
$89$ $$142 + T$$
$97$ $$1407 + T$$