# Properties

 Label 20.4.e.a Level 20 Weight 4 Character orbit 20.e Analytic conductor 1.180 Analytic rank 0 Dimension 2 CM disc. -4 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.18003820011$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 2 + 2 i ) q^{2}$$ $$+ 8 i q^{4}$$ $$+ ( -2 - 11 i ) q^{5}$$ $$+ ( -16 + 16 i ) q^{8}$$ $$-27 i q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 2 + 2 i ) q^{2}$$ $$+ 8 i q^{4}$$ $$+ ( -2 - 11 i ) q^{5}$$ $$+ ( -16 + 16 i ) q^{8}$$ $$-27 i q^{9}$$ $$+ ( 18 - 26 i ) q^{10}$$ $$+ ( -37 + 37 i ) q^{13}$$ $$-64 q^{16}$$ $$+ ( 99 + 99 i ) q^{17}$$ $$+ ( 54 - 54 i ) q^{18}$$ $$+ ( 88 - 16 i ) q^{20}$$ $$+ ( -117 + 44 i ) q^{25}$$ $$-148 q^{26}$$ $$-284 i q^{29}$$ $$+ ( -128 - 128 i ) q^{32}$$ $$+ 396 i q^{34}$$ $$+ 216 q^{36}$$ $$+ ( -91 - 91 i ) q^{37}$$ $$+ ( 208 + 144 i ) q^{40}$$ $$+ 472 q^{41}$$ $$+ ( -297 + 54 i ) q^{45}$$ $$+ 343 i q^{49}$$ $$+ ( -322 - 146 i ) q^{50}$$ $$+ ( -296 - 296 i ) q^{52}$$ $$+ ( -27 + 27 i ) q^{53}$$ $$+ ( 568 - 568 i ) q^{58}$$ $$-468 q^{61}$$ $$-512 i q^{64}$$ $$+ ( 481 + 333 i ) q^{65}$$ $$+ ( -792 + 792 i ) q^{68}$$ $$+ ( 432 + 432 i ) q^{72}$$ $$+ ( 253 - 253 i ) q^{73}$$ $$-364 i q^{74}$$ $$+ ( 128 + 704 i ) q^{80}$$ $$-729 q^{81}$$ $$+ ( 944 + 944 i ) q^{82}$$ $$+ ( 891 - 1287 i ) q^{85}$$ $$+ 176 i q^{89}$$ $$+ ( -702 - 486 i ) q^{90}$$ $$+ ( -611 - 611 i ) q^{97}$$ $$+ ( -686 + 686 i ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 4q^{5}$$ $$\mathstrut -\mathstrut 32q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 4q^{5}$$ $$\mathstrut -\mathstrut 32q^{8}$$ $$\mathstrut +\mathstrut 36q^{10}$$ $$\mathstrut -\mathstrut 74q^{13}$$ $$\mathstrut -\mathstrut 128q^{16}$$ $$\mathstrut +\mathstrut 198q^{17}$$ $$\mathstrut +\mathstrut 108q^{18}$$ $$\mathstrut +\mathstrut 176q^{20}$$ $$\mathstrut -\mathstrut 234q^{25}$$ $$\mathstrut -\mathstrut 296q^{26}$$ $$\mathstrut -\mathstrut 256q^{32}$$ $$\mathstrut +\mathstrut 432q^{36}$$ $$\mathstrut -\mathstrut 182q^{37}$$ $$\mathstrut +\mathstrut 416q^{40}$$ $$\mathstrut +\mathstrut 944q^{41}$$ $$\mathstrut -\mathstrut 594q^{45}$$ $$\mathstrut -\mathstrut 644q^{50}$$ $$\mathstrut -\mathstrut 592q^{52}$$ $$\mathstrut -\mathstrut 54q^{53}$$ $$\mathstrut +\mathstrut 1136q^{58}$$ $$\mathstrut -\mathstrut 936q^{61}$$ $$\mathstrut +\mathstrut 962q^{65}$$ $$\mathstrut -\mathstrut 1584q^{68}$$ $$\mathstrut +\mathstrut 864q^{72}$$ $$\mathstrut +\mathstrut 506q^{73}$$ $$\mathstrut +\mathstrut 256q^{80}$$ $$\mathstrut -\mathstrut 1458q^{81}$$ $$\mathstrut +\mathstrut 1888q^{82}$$ $$\mathstrut +\mathstrut 1782q^{85}$$ $$\mathstrut -\mathstrut 1404q^{90}$$ $$\mathstrut -\mathstrut 1222q^{97}$$ $$\mathstrut -\mathstrut 1372q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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
2.00000 2.00000i 0 8.00000i −2.00000 + 11.0000i 0 0 −16.0000 16.0000i 27.0000i 18.0000 + 26.0000i
7.1 2.00000 + 2.00000i 0 8.00000i −2.00000 11.0000i 0 0 −16.0000 + 16.0000i 27.0000i 18.0000 26.0000i
 $$n$$: 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

This newform can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{4}^{\mathrm{new}}(20, [\chi])$$.