# Properties

 Label 108.4.a.c Level 108 Weight 4 Character orbit 108.a Self dual yes Analytic conductor 6.372 Analytic rank 0 Dimension 1 CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 108.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.37220628062$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 17q^{7} + O(q^{10})$$ $$q + 17q^{7} + 89q^{13} + 107q^{19} - 125q^{25} + 308q^{31} - 433q^{37} - 520q^{43} - 54q^{49} - 901q^{61} + 1007q^{67} - 271q^{73} + 503q^{79} + 1513q^{91} + 1853q^{97} + O(q^{100})$$

## 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
0 0 0 0 0 17.0000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.a.c 1
3.b odd 2 1 CM 108.4.a.c 1
4.b odd 2 1 432.4.a.g 1
8.b even 2 1 1728.4.a.q 1
8.d odd 2 1 1728.4.a.p 1
9.c even 3 2 324.4.e.d 2
9.d odd 6 2 324.4.e.d 2
12.b even 2 1 432.4.a.g 1
24.f even 2 1 1728.4.a.p 1
24.h odd 2 1 1728.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.c 1 1.a even 1 1 trivial
108.4.a.c 1 3.b odd 2 1 CM
324.4.e.d 2 9.c even 3 2
324.4.e.d 2 9.d odd 6 2
432.4.a.g 1 4.b odd 2 1
432.4.a.g 1 12.b even 2 1
1728.4.a.p 1 8.d odd 2 1
1728.4.a.p 1 24.f even 2 1
1728.4.a.q 1 8.b even 2 1
1728.4.a.q 1 24.h odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(108))$$:

 $$T_{5}$$ $$T_{7} - 17$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 125 T^{2}$$
$7$ $$1 - 17 T + 343 T^{2}$$
$11$ $$1 + 1331 T^{2}$$
$13$ $$1 - 89 T + 2197 T^{2}$$
$17$ $$1 + 4913 T^{2}$$
$19$ $$1 - 107 T + 6859 T^{2}$$
$23$ $$1 + 12167 T^{2}$$
$29$ $$1 + 24389 T^{2}$$
$31$ $$1 - 308 T + 29791 T^{2}$$
$37$ $$1 + 433 T + 50653 T^{2}$$
$41$ $$1 + 68921 T^{2}$$
$43$ $$1 + 520 T + 79507 T^{2}$$
$47$ $$1 + 103823 T^{2}$$
$53$ $$1 + 148877 T^{2}$$
$59$ $$1 + 205379 T^{2}$$
$61$ $$1 + 901 T + 226981 T^{2}$$
$67$ $$1 - 1007 T + 300763 T^{2}$$
$71$ $$1 + 357911 T^{2}$$
$73$ $$1 + 271 T + 389017 T^{2}$$
$79$ $$1 - 503 T + 493039 T^{2}$$
$83$ $$1 + 571787 T^{2}$$
$89$ $$1 + 704969 T^{2}$$
$97$ $$1 - 1853 T + 912673 T^{2}$$