# Properties

 Label 120.1.i.a Level 120 Weight 1 Character orbit 120.i Analytic conductor 0.060 Analytic rank 0 Dimension 2 Projective image $$D_{2}$$ CM/RM disc. -15, -24, 40 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 120.i (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.059887801516$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-6}, \sqrt{10})$$ Artin image size $$16$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.3240000.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q$$ $$-i q^{2}$$ $$-i q^{3}$$ $$- q^{4}$$ $$+ i q^{5}$$ $$- q^{6}$$ $$+ i q^{8}$$ $$- q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-i q^{2}$$ $$-i q^{3}$$ $$- q^{4}$$ $$+ i q^{5}$$ $$- q^{6}$$ $$+ i q^{8}$$ $$- q^{9}$$ $$+ q^{10}$$ $$+ i q^{12}$$ $$+ q^{15}$$ $$+ q^{16}$$ $$+ i q^{18}$$ $$-i q^{20}$$ $$+ q^{24}$$ $$- q^{25}$$ $$+ i q^{27}$$ $$-i q^{30}$$ $$-2 q^{31}$$ $$-i q^{32}$$ $$+ q^{36}$$ $$- q^{40}$$ $$-i q^{45}$$ $$-i q^{48}$$ $$+ q^{49}$$ $$+ i q^{50}$$ $$-2 i q^{53}$$ $$+ q^{54}$$ $$- q^{60}$$ $$+ 2 i q^{62}$$ $$- q^{64}$$ $$-i q^{72}$$ $$+ i q^{75}$$ $$+ 2 q^{79}$$ $$+ i q^{80}$$ $$+ q^{81}$$ $$+ 2 i q^{83}$$ $$- q^{90}$$ $$+ 2 i q^{93}$$ $$- q^{96}$$ $$-i q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 2q^{4}$$ $$\mathstrut -\mathstrut 2q^{6}$$ $$\mathstrut -\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 2q^{4}$$ $$\mathstrut -\mathstrut 2q^{6}$$ $$\mathstrut -\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut 2q^{10}$$ $$\mathstrut +\mathstrut 2q^{15}$$ $$\mathstrut +\mathstrut 2q^{16}$$ $$\mathstrut +\mathstrut 2q^{24}$$ $$\mathstrut -\mathstrut 2q^{25}$$ $$\mathstrut -\mathstrut 4q^{31}$$ $$\mathstrut +\mathstrut 2q^{36}$$ $$\mathstrut -\mathstrut 2q^{40}$$ $$\mathstrut +\mathstrut 2q^{49}$$ $$\mathstrut +\mathstrut 2q^{54}$$ $$\mathstrut -\mathstrut 2q^{60}$$ $$\mathstrut -\mathstrut 2q^{64}$$ $$\mathstrut +\mathstrut 4q^{79}$$ $$\mathstrut +\mathstrut 2q^{81}$$ $$\mathstrut -\mathstrut 2q^{90}$$ $$\mathstrut -\mathstrut 2q^{96}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$41$$ $$61$$ $$97$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 1.00000i −1.00000 0 1.00000i −1.00000 1.00000
29.2 1.00000i 1.00000i −1.00000 1.00000i −1.00000 0 1.00000i −1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
15.d Odd 1 CM by $$\Q(\sqrt{-15})$$ yes
24.h Odd 1 CM by $$\Q(\sqrt{-6})$$ yes
40.f Even 1 RM by $$\Q(\sqrt{10})$$ yes
3.b Odd 1 yes
5.b Even 1 yes
8.b Even 1 yes
120.i Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{1}^{\mathrm{new}}(120, [\chi])$$.