# Properties

 Label 20.4.e.a Level 20 Weight 4 Character orbit 20.e Analytic conductor 1.180 Analytic rank 0 Dimension 2 CM discriminant -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 $$4$$, degree $$2$$, minimal)

## 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: $$1$$ Twist minimal: yes 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.

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

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.4.e.a 2
3.b odd 2 1 180.4.k.a 2
4.b odd 2 1 CM 20.4.e.a 2
5.b even 2 1 100.4.e.a 2
5.c odd 4 1 inner 20.4.e.a 2
5.c odd 4 1 100.4.e.a 2
8.b even 2 1 320.4.n.c 2
8.d odd 2 1 320.4.n.c 2
12.b even 2 1 180.4.k.a 2
15.e even 4 1 180.4.k.a 2
20.d odd 2 1 100.4.e.a 2
20.e even 4 1 inner 20.4.e.a 2
20.e even 4 1 100.4.e.a 2
40.i odd 4 1 320.4.n.c 2
40.k even 4 1 320.4.n.c 2
60.l odd 4 1 180.4.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.e.a 2 1.a even 1 1 trivial
20.4.e.a 2 4.b odd 2 1 CM
20.4.e.a 2 5.c odd 4 1 inner
20.4.e.a 2 20.e even 4 1 inner
100.4.e.a 2 5.b even 2 1
100.4.e.a 2 5.c odd 4 1
100.4.e.a 2 20.d odd 2 1
100.4.e.a 2 20.e even 4 1
180.4.k.a 2 3.b odd 2 1
180.4.k.a 2 12.b even 2 1
180.4.k.a 2 15.e even 4 1
180.4.k.a 2 60.l odd 4 1
320.4.n.c 2 8.b even 2 1
320.4.n.c 2 8.d odd 2 1
320.4.n.c 2 40.i odd 4 1
320.4.n.c 2 40.k even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 8 T^{2}$$
$3$ $$1 + 729 T^{4}$$
$5$ $$1 + 4 T + 125 T^{2}$$
$7$ $$1 + 117649 T^{4}$$
$11$ $$( 1 - 1331 T^{2} )^{2}$$
$13$ $$( 1 - 18 T + 2197 T^{2} )( 1 + 92 T + 2197 T^{2} )$$
$17$ $$( 1 - 104 T + 4913 T^{2} )( 1 - 94 T + 4913 T^{2} )$$
$19$ $$( 1 + 6859 T^{2} )^{2}$$
$23$ $$1 + 148035889 T^{4}$$
$29$ $$( 1 - 130 T + 24389 T^{2} )( 1 + 130 T + 24389 T^{2} )$$
$31$ $$( 1 - 29791 T^{2} )^{2}$$
$37$ $$( 1 - 214 T + 50653 T^{2} )( 1 + 396 T + 50653 T^{2} )$$
$41$ $$( 1 - 472 T + 68921 T^{2} )^{2}$$
$43$ $$1 + 6321363049 T^{4}$$
$47$ $$1 + 10779215329 T^{4}$$
$53$ $$( 1 - 518 T + 148877 T^{2} )( 1 + 572 T + 148877 T^{2} )$$
$59$ $$( 1 + 205379 T^{2} )^{2}$$
$61$ $$( 1 + 468 T + 226981 T^{2} )^{2}$$
$67$ $$1 + 90458382169 T^{4}$$
$71$ $$( 1 - 357911 T^{2} )^{2}$$
$73$ $$( 1 - 1098 T + 389017 T^{2} )( 1 + 592 T + 389017 T^{2} )$$
$79$ $$( 1 + 493039 T^{2} )^{2}$$
$83$ $$1 + 326940373369 T^{4}$$
$89$ $$( 1 - 1670 T + 704969 T^{2} )( 1 + 1670 T + 704969 T^{2} )$$
$97$ $$( 1 - 594 T + 912673 T^{2} )( 1 + 1816 T + 912673 T^{2} )$$