# Properties

 Label 20.2.e.a Level 20 Weight 2 Character orbit 20.e Analytic conductor 0.160 Analytic rank 0 Dimension 2 CM disc. -4 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 20.e (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.159700804043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 - i ) q^{2}$$ $$+ 2 i q^{4}$$ $$+ ( -2 + i ) q^{5}$$ $$+ ( 2 - 2 i ) q^{8}$$ $$-3 i q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 - i ) q^{2}$$ $$+ 2 i q^{4}$$ $$+ ( -2 + i ) q^{5}$$ $$+ ( 2 - 2 i ) q^{8}$$ $$-3 i q^{9}$$ $$+ ( 3 + i ) q^{10}$$ $$+ ( -1 + i ) q^{13}$$ $$-4 q^{16}$$ $$+ ( 3 + 3 i ) q^{17}$$ $$+ ( -3 + 3 i ) q^{18}$$ $$+ ( -2 - 4 i ) q^{20}$$ $$+ ( 3 - 4 i ) q^{25}$$ $$+ 2 q^{26}$$ $$+ 4 i q^{29}$$ $$+ ( 4 + 4 i ) q^{32}$$ $$-6 i q^{34}$$ $$+ 6 q^{36}$$ $$+ ( -7 - 7 i ) q^{37}$$ $$+ ( -2 + 6 i ) q^{40}$$ $$-8 q^{41}$$ $$+ ( 3 + 6 i ) q^{45}$$ $$+ 7 i q^{49}$$ $$+ ( -7 + i ) q^{50}$$ $$+ ( -2 - 2 i ) q^{52}$$ $$+ ( 9 - 9 i ) q^{53}$$ $$+ ( 4 - 4 i ) q^{58}$$ $$+ 12 q^{61}$$ $$-8 i q^{64}$$ $$+ ( 1 - 3 i ) q^{65}$$ $$+ ( -6 + 6 i ) q^{68}$$ $$+ ( -6 - 6 i ) q^{72}$$ $$+ ( -11 + 11 i ) q^{73}$$ $$+ 14 i q^{74}$$ $$+ ( 8 - 4 i ) q^{80}$$ $$-9 q^{81}$$ $$+ ( 8 + 8 i ) q^{82}$$ $$+ ( -9 - 3 i ) q^{85}$$ $$-16 i q^{89}$$ $$+ ( 3 - 9 i ) q^{90}$$ $$+ ( 13 + 13 i ) q^{97}$$ $$+ ( 7 - 7 i ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{5}$$ $$\mathstrut +\mathstrut 4q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{5}$$ $$\mathstrut +\mathstrut 4q^{8}$$ $$\mathstrut +\mathstrut 6q^{10}$$ $$\mathstrut -\mathstrut 2q^{13}$$ $$\mathstrut -\mathstrut 8q^{16}$$ $$\mathstrut +\mathstrut 6q^{17}$$ $$\mathstrut -\mathstrut 6q^{18}$$ $$\mathstrut -\mathstrut 4q^{20}$$ $$\mathstrut +\mathstrut 6q^{25}$$ $$\mathstrut +\mathstrut 4q^{26}$$ $$\mathstrut +\mathstrut 8q^{32}$$ $$\mathstrut +\mathstrut 12q^{36}$$ $$\mathstrut -\mathstrut 14q^{37}$$ $$\mathstrut -\mathstrut 4q^{40}$$ $$\mathstrut -\mathstrut 16q^{41}$$ $$\mathstrut +\mathstrut 6q^{45}$$ $$\mathstrut -\mathstrut 14q^{50}$$ $$\mathstrut -\mathstrut 4q^{52}$$ $$\mathstrut +\mathstrut 18q^{53}$$ $$\mathstrut +\mathstrut 8q^{58}$$ $$\mathstrut +\mathstrut 24q^{61}$$ $$\mathstrut +\mathstrut 2q^{65}$$ $$\mathstrut -\mathstrut 12q^{68}$$ $$\mathstrut -\mathstrut 12q^{72}$$ $$\mathstrut -\mathstrut 22q^{73}$$ $$\mathstrut +\mathstrut 16q^{80}$$ $$\mathstrut -\mathstrut 18q^{81}$$ $$\mathstrut +\mathstrut 16q^{82}$$ $$\mathstrut -\mathstrut 18q^{85}$$ $$\mathstrut +\mathstrut 6q^{90}$$ $$\mathstrut +\mathstrut 26q^{97}$$ $$\mathstrut +\mathstrut 14q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$-1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 1.00000i 1.00000i
−1.00000 + 1.00000i 0 2.00000i −2.00000 1.00000i 0 0 2.00000 + 2.00000i 3.00000i 3.00000 1.00000i
7.1 −1.00000 1.00000i 0 2.00000i −2.00000 + 1.00000i 0 0 2.00000 2.00000i 3.00000i 3.00000 + 1.00000i
 $$n$$: 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
4.b Odd 1 CM by $$\Q(\sqrt{-1})$$ yes
5.c Odd 1 yes
20.e Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(20, [\chi])$$.