# Properties

 Label 20.10.e.a Level 20 Weight 10 Character orbit 20.e Analytic conductor 10.301 Analytic rank 0 Dimension 2 CM disc. -4 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 20.e (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$10.3007167233$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -16 - 16 i ) q^{2}$$ $$+ 512 i q^{4}$$ $$+ ( 718 - 1199 i ) q^{5}$$ $$+ ( 8192 - 8192 i ) q^{8}$$ $$-19683 i q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -16 - 16 i ) q^{2}$$ $$+ 512 i q^{4}$$ $$+ ( 718 - 1199 i ) q^{5}$$ $$+ ( 8192 - 8192 i ) q^{8}$$ $$-19683 i q^{9}$$ $$+ ( -30672 + 7696 i ) q^{10}$$ $$+ ( -142561 + 142561 i ) q^{13}$$ $$-262144 q^{16}$$ $$+ ( -481437 - 481437 i ) q^{17}$$ $$+ ( -314928 + 314928 i ) q^{18}$$ $$+ ( 613888 + 367616 i ) q^{20}$$ $$+ ( -922077 - 1721764 i ) q^{25}$$ $$+ 4561952 q^{26}$$ $$-2126876 i q^{29}$$ $$+ ( 4194304 + 4194304 i ) q^{32}$$ $$+ 15405984 i q^{34}$$ $$+ 10077696 q^{36}$$ $$+ ( -12321127 - 12321127 i ) q^{37}$$ $$+ ( -3940352 - 15704064 i ) q^{40}$$ $$+ 7561912 q^{41}$$ $$+ ( -23599917 - 14132394 i ) q^{45}$$ $$+ 40353607 i q^{49}$$ $$+ ( -12794992 + 42301456 i ) q^{50}$$ $$+ ( -72991232 - 72991232 i ) q^{52}$$ $$+ ( -12019671 + 12019671 i ) q^{53}$$ $$+ ( -34030016 + 34030016 i ) q^{58}$$ $$+ 216178092 q^{61}$$ $$-134217728 i q^{64}$$ $$+ ( 68571841 + 273289437 i ) q^{65}$$ $$+ ( 246495744 - 246495744 i ) q^{68}$$ $$+ ( -161243136 - 161243136 i ) q^{72}$$ $$+ ( 262875349 - 262875349 i ) q^{73}$$ $$+ 394276064 i q^{74}$$ $$+ ( -188219392 + 314310656 i ) q^{80}$$ $$-387420489 q^{81}$$ $$+ ( -120990592 - 120990592 i ) q^{82}$$ $$+ ( -922914729 + 231571197 i ) q^{85}$$ $$-366771856 i q^{89}$$ $$+ ( 151480368 + 603716976 i ) q^{90}$$ $$+ ( -1216731347 - 1216731347 i ) q^{97}$$ $$+ ( 645657712 - 645657712 i ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 32q^{2}$$ $$\mathstrut +\mathstrut 1436q^{5}$$ $$\mathstrut +\mathstrut 16384q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 32q^{2}$$ $$\mathstrut +\mathstrut 1436q^{5}$$ $$\mathstrut +\mathstrut 16384q^{8}$$ $$\mathstrut -\mathstrut 61344q^{10}$$ $$\mathstrut -\mathstrut 285122q^{13}$$ $$\mathstrut -\mathstrut 524288q^{16}$$ $$\mathstrut -\mathstrut 962874q^{17}$$ $$\mathstrut -\mathstrut 629856q^{18}$$ $$\mathstrut +\mathstrut 1227776q^{20}$$ $$\mathstrut -\mathstrut 1844154q^{25}$$ $$\mathstrut +\mathstrut 9123904q^{26}$$ $$\mathstrut +\mathstrut 8388608q^{32}$$ $$\mathstrut +\mathstrut 20155392q^{36}$$ $$\mathstrut -\mathstrut 24642254q^{37}$$ $$\mathstrut -\mathstrut 7880704q^{40}$$ $$\mathstrut +\mathstrut 15123824q^{41}$$ $$\mathstrut -\mathstrut 47199834q^{45}$$ $$\mathstrut -\mathstrut 25589984q^{50}$$ $$\mathstrut -\mathstrut 145982464q^{52}$$ $$\mathstrut -\mathstrut 24039342q^{53}$$ $$\mathstrut -\mathstrut 68060032q^{58}$$ $$\mathstrut +\mathstrut 432356184q^{61}$$ $$\mathstrut +\mathstrut 137143682q^{65}$$ $$\mathstrut +\mathstrut 492991488q^{68}$$ $$\mathstrut -\mathstrut 322486272q^{72}$$ $$\mathstrut +\mathstrut 525750698q^{73}$$ $$\mathstrut -\mathstrut 376438784q^{80}$$ $$\mathstrut -\mathstrut 774840978q^{81}$$ $$\mathstrut -\mathstrut 241981184q^{82}$$ $$\mathstrut -\mathstrut 1845829458q^{85}$$ $$\mathstrut +\mathstrut 302960736q^{90}$$ $$\mathstrut -\mathstrut 2433462694q^{97}$$ $$\mathstrut +\mathstrut 1291315424q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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
−16.0000 + 16.0000i 0 512.000i 718.000 + 1199.00i 0 0 8192.00 + 8192.00i 19683.0i −30672.0 7696.00i
7.1 −16.0000 16.0000i 0 512.000i 718.000 1199.00i 0 0 8192.00 8192.00i 19683.0i −30672.0 + 7696.00i
 $$n$$: 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 $$S_{10}^{\mathrm{new}}(20, [\chi])$$.