# Properties

 Label 20.4.c.a Level 20 Weight 4 Character orbit 20.c Analytic conductor 1.180 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 20.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.18003820011$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + ( 7 + \beta ) q^{5} + \beta q^{7} -49 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + ( 7 + \beta ) q^{5} + \beta q^{7} -49 q^{9} + 20 q^{11} + 6 \beta q^{13} + ( 76 - 7 \beta ) q^{15} -8 \beta q^{17} -84 q^{19} + 76 q^{21} -7 \beta q^{23} + ( -27 + 14 \beta ) q^{25} + 22 \beta q^{27} + 6 q^{29} -224 q^{31} -20 \beta q^{33} + ( -76 + 7 \beta ) q^{35} -14 \beta q^{37} + 456 q^{39} + 266 q^{41} + 35 \beta q^{43} + ( -343 - 49 \beta ) q^{45} -43 \beta q^{47} + 267 q^{49} -608 q^{51} + 42 \beta q^{53} + ( 140 + 20 \beta ) q^{55} + 84 \beta q^{57} -28 q^{59} + 182 q^{61} -49 \beta q^{63} + ( -456 + 42 \beta ) q^{65} -49 \beta q^{67} -532 q^{69} + 408 q^{71} -124 \beta q^{73} + ( 1064 + 27 \beta ) q^{75} + 20 \beta q^{77} + 48 q^{79} + 349 q^{81} + 23 \beta q^{83} + ( 608 - 56 \beta ) q^{85} -6 \beta q^{87} -1526 q^{89} -456 q^{91} + 224 \beta q^{93} + ( -588 - 84 \beta ) q^{95} -64 \beta q^{97} -980 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{5} - 98q^{9} + O(q^{10})$$ $$2q + 14q^{5} - 98q^{9} + 40q^{11} + 152q^{15} - 168q^{19} + 152q^{21} - 54q^{25} + 12q^{29} - 448q^{31} - 152q^{35} + 912q^{39} + 532q^{41} - 686q^{45} + 534q^{49} - 1216q^{51} + 280q^{55} - 56q^{59} + 364q^{61} - 912q^{65} - 1064q^{69} + 816q^{71} + 2128q^{75} + 96q^{79} + 698q^{81} + 1216q^{85} - 3052q^{89} - 912q^{91} - 1176q^{95} - 1960q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.5 + 2.17945i 0.5 − 2.17945i
0 8.71780i 0 7.00000 + 8.71780i 0 8.71780i 0 −49.0000 0
9.2 0 8.71780i 0 7.00000 8.71780i 0 8.71780i 0 −49.0000 0
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.4.c.a 2
3.b odd 2 1 180.4.d.a 2
4.b odd 2 1 80.4.c.b 2
5.b even 2 1 inner 20.4.c.a 2
5.c odd 4 2 100.4.a.d 2
7.b odd 2 1 980.4.e.a 2
8.b even 2 1 320.4.c.a 2
8.d odd 2 1 320.4.c.b 2
12.b even 2 1 720.4.f.a 2
15.d odd 2 1 180.4.d.a 2
15.e even 4 2 900.4.a.s 2
20.d odd 2 1 80.4.c.b 2
20.e even 4 2 400.4.a.w 2
35.c odd 2 1 980.4.e.a 2
40.e odd 2 1 320.4.c.b 2
40.f even 2 1 320.4.c.a 2
40.i odd 4 2 1600.4.a.cj 2
40.k even 4 2 1600.4.a.ck 2
60.h even 2 1 720.4.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.c.a 2 1.a even 1 1 trivial
20.4.c.a 2 5.b even 2 1 inner
80.4.c.b 2 4.b odd 2 1
80.4.c.b 2 20.d odd 2 1
100.4.a.d 2 5.c odd 4 2
180.4.d.a 2 3.b odd 2 1
180.4.d.a 2 15.d odd 2 1
320.4.c.a 2 8.b even 2 1
320.4.c.a 2 40.f even 2 1
320.4.c.b 2 8.d odd 2 1
320.4.c.b 2 40.e odd 2 1
400.4.a.w 2 20.e even 4 2
720.4.f.a 2 12.b even 2 1
720.4.f.a 2 60.h even 2 1
900.4.a.s 2 15.e even 4 2
980.4.e.a 2 7.b odd 2 1
980.4.e.a 2 35.c odd 2 1
1600.4.a.cj 2 40.i odd 4 2
1600.4.a.ck 2 40.k even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(20, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 22 T^{2} + 729 T^{4}$$
$5$ $$1 - 14 T + 125 T^{2}$$
$7$ $$( 1 - 36 T + 343 T^{2} )( 1 + 36 T + 343 T^{2} )$$
$11$ $$( 1 - 20 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 1658 T^{2} + 4826809 T^{4}$$
$17$ $$1 - 4962 T^{2} + 24137569 T^{4}$$
$19$ $$( 1 + 84 T + 6859 T^{2} )^{2}$$
$23$ $$( 1 - 212 T + 12167 T^{2} )( 1 + 212 T + 12167 T^{2} )$$
$29$ $$( 1 - 6 T + 24389 T^{2} )^{2}$$
$31$ $$( 1 + 224 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 86410 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 - 266 T + 68921 T^{2} )^{2}$$
$43$ $$1 - 65914 T^{2} + 6321363049 T^{4}$$
$47$ $$1 - 67122 T^{2} + 10779215329 T^{4}$$
$53$ $$1 - 163690 T^{2} + 22164361129 T^{4}$$
$59$ $$( 1 + 28 T + 205379 T^{2} )^{2}$$
$61$ $$( 1 - 182 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 419050 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 - 408 T + 357911 T^{2} )^{2}$$
$73$ $$1 + 390542 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 - 48 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 1103370 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 + 1526 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 1514050 T^{2} + 832972004929 T^{4}$$