Properties

 Label 20.3.d.c Level 20 Weight 3 Character orbit 20.d Analytic conductor 0.545 Analytic rank 0 Dimension 2 CM discriminant -4 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 20.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.544960528721$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} -4 q^{4} + ( 3 - 4 i ) q^{5} -8 i q^{8} -9 q^{9} +O(q^{10})$$ $$q + 2 i q^{2} -4 q^{4} + ( 3 - 4 i ) q^{5} -8 i q^{8} -9 q^{9} + ( 8 + 6 i ) q^{10} + 24 i q^{13} + 16 q^{16} -16 i q^{17} -18 i q^{18} + ( -12 + 16 i ) q^{20} + ( -7 - 24 i ) q^{25} -48 q^{26} + 42 q^{29} + 32 i q^{32} + 32 q^{34} + 36 q^{36} + 24 i q^{37} + ( -32 - 24 i ) q^{40} -18 q^{41} + ( -27 + 36 i ) q^{45} -49 q^{49} + ( 48 - 14 i ) q^{50} -96 i q^{52} -56 i q^{53} + 84 i q^{58} + 22 q^{61} -64 q^{64} + ( 96 + 72 i ) q^{65} + 64 i q^{68} + 72 i q^{72} -96 i q^{73} -48 q^{74} + ( 48 - 64 i ) q^{80} + 81 q^{81} -36 i q^{82} + ( -64 - 48 i ) q^{85} -78 q^{89} + ( -72 - 54 i ) q^{90} + 144 i q^{97} -98 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} + 6q^{5} - 18q^{9} + O(q^{10})$$ $$2q - 8q^{4} + 6q^{5} - 18q^{9} + 16q^{10} + 32q^{16} - 24q^{20} - 14q^{25} - 96q^{26} + 84q^{29} + 64q^{34} + 72q^{36} - 64q^{40} - 36q^{41} - 54q^{45} - 98q^{49} + 96q^{50} + 44q^{61} - 128q^{64} + 192q^{65} - 96q^{74} + 96q^{80} + 162q^{81} - 128q^{85} - 156q^{89} - 144q^{90} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 1.00000i 1.00000i
2.00000i 0 −4.00000 3.00000 + 4.00000i 0 0 8.00000i −9.00000 8.00000 6.00000i
19.2 2.00000i 0 −4.00000 3.00000 4.00000i 0 0 8.00000i −9.00000 8.00000 + 6.00000i
 $$n$$: 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.b even 2 1 inner
20.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.3.d.c 2
3.b odd 2 1 180.3.f.c 2
4.b odd 2 1 CM 20.3.d.c 2
5.b even 2 1 inner 20.3.d.c 2
5.c odd 4 1 100.3.b.a 1
5.c odd 4 1 100.3.b.b 1
8.b even 2 1 320.3.h.d 2
8.d odd 2 1 320.3.h.d 2
12.b even 2 1 180.3.f.c 2
15.d odd 2 1 180.3.f.c 2
15.e even 4 1 900.3.c.a 1
15.e even 4 1 900.3.c.d 1
16.e even 4 1 1280.3.e.a 2
16.e even 4 1 1280.3.e.d 2
16.f odd 4 1 1280.3.e.a 2
16.f odd 4 1 1280.3.e.d 2
20.d odd 2 1 inner 20.3.d.c 2
20.e even 4 1 100.3.b.a 1
20.e even 4 1 100.3.b.b 1
40.e odd 2 1 320.3.h.d 2
40.f even 2 1 320.3.h.d 2
40.i odd 4 1 1600.3.b.a 1
40.i odd 4 1 1600.3.b.c 1
40.k even 4 1 1600.3.b.a 1
40.k even 4 1 1600.3.b.c 1
60.h even 2 1 180.3.f.c 2
60.l odd 4 1 900.3.c.a 1
60.l odd 4 1 900.3.c.d 1
80.k odd 4 1 1280.3.e.a 2
80.k odd 4 1 1280.3.e.d 2
80.q even 4 1 1280.3.e.a 2
80.q even 4 1 1280.3.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.c 2 1.a even 1 1 trivial
20.3.d.c 2 4.b odd 2 1 CM
20.3.d.c 2 5.b even 2 1 inner
20.3.d.c 2 20.d odd 2 1 inner
100.3.b.a 1 5.c odd 4 1
100.3.b.a 1 20.e even 4 1
100.3.b.b 1 5.c odd 4 1
100.3.b.b 1 20.e even 4 1
180.3.f.c 2 3.b odd 2 1
180.3.f.c 2 12.b even 2 1
180.3.f.c 2 15.d odd 2 1
180.3.f.c 2 60.h even 2 1
320.3.h.d 2 8.b even 2 1
320.3.h.d 2 8.d odd 2 1
320.3.h.d 2 40.e odd 2 1
320.3.h.d 2 40.f even 2 1
900.3.c.a 1 15.e even 4 1
900.3.c.a 1 60.l odd 4 1
900.3.c.d 1 15.e even 4 1
900.3.c.d 1 60.l odd 4 1
1280.3.e.a 2 16.e even 4 1
1280.3.e.a 2 16.f odd 4 1
1280.3.e.a 2 80.k odd 4 1
1280.3.e.a 2 80.q even 4 1
1280.3.e.d 2 16.e even 4 1
1280.3.e.d 2 16.f odd 4 1
1280.3.e.d 2 80.k odd 4 1
1280.3.e.d 2 80.q even 4 1
1600.3.b.a 1 40.i odd 4 1
1600.3.b.a 1 40.k even 4 1
1600.3.b.c 1 40.i odd 4 1
1600.3.b.c 1 40.k even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2}$$
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ $$1 - 6 T + 25 T^{2}$$
$7$ $$( 1 + 49 T^{2} )^{2}$$
$11$ $$( 1 - 11 T )^{2}( 1 + 11 T )^{2}$$
$13$ $$( 1 - 10 T + 169 T^{2} )( 1 + 10 T + 169 T^{2} )$$
$17$ $$( 1 - 30 T + 289 T^{2} )( 1 + 30 T + 289 T^{2} )$$
$19$ $$( 1 - 19 T )^{2}( 1 + 19 T )^{2}$$
$23$ $$( 1 + 529 T^{2} )^{2}$$
$29$ $$( 1 - 42 T + 841 T^{2} )^{2}$$
$31$ $$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$
$37$ $$( 1 - 70 T + 1369 T^{2} )( 1 + 70 T + 1369 T^{2} )$$
$41$ $$( 1 + 18 T + 1681 T^{2} )^{2}$$
$43$ $$( 1 + 1849 T^{2} )^{2}$$
$47$ $$( 1 + 2209 T^{2} )^{2}$$
$53$ $$( 1 - 90 T + 2809 T^{2} )( 1 + 90 T + 2809 T^{2} )$$
$59$ $$( 1 - 59 T )^{2}( 1 + 59 T )^{2}$$
$61$ $$( 1 - 22 T + 3721 T^{2} )^{2}$$
$67$ $$( 1 + 4489 T^{2} )^{2}$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 - 110 T + 5329 T^{2} )( 1 + 110 T + 5329 T^{2} )$$
$79$ $$( 1 - 79 T )^{2}( 1 + 79 T )^{2}$$
$83$ $$( 1 + 6889 T^{2} )^{2}$$
$89$ $$( 1 + 78 T + 7921 T^{2} )^{2}$$
$97$ $$( 1 - 130 T + 9409 T^{2} )( 1 + 130 T + 9409 T^{2} )$$