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

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,4,Mod(9,20)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("20.9");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.18003820011$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 5$$ x^2 - x + 5 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 76$$
$5$ $$T^{2} - 14T + 125$$
$7$ $$T^{2} + 76$$
$11$ $$(T - 20)^{2}$$
$13$ $$T^{2} + 2736$$
$17$ $$T^{2} + 4864$$
$19$ $$(T + 84)^{2}$$
$23$ $$T^{2} + 3724$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 224)^{2}$$
$37$ $$T^{2} + 14896$$
$41$ $$(T - 266)^{2}$$
$43$ $$T^{2} + 93100$$
$47$ $$T^{2} + 140524$$
$53$ $$T^{2} + 134064$$
$59$ $$(T + 28)^{2}$$
$61$ $$(T - 182)^{2}$$
$67$ $$T^{2} + 182476$$
$71$ $$(T - 408)^{2}$$
$73$ $$T^{2} + 1168576$$
$79$ $$(T - 48)^{2}$$
$83$ $$T^{2} + 40204$$
$89$ $$(T + 1526)^{2}$$
$97$ $$T^{2} + 311296$$