# Properties

 Label 20.3.b.a Level $20$ Weight $3$ Character orbit 20.b Analytic conductor $0.545$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,3,Mod(11,20)]

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([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("20.11");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 20.b (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: $$0.544960528721$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{6} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{7} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{8} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + b3 * q^3 + (-b3 + b2 - b1 - 2) * q^4 + (b2 - b1 - 1) * q^5 + (-b3 - b2 + 3*b1 + 2) * q^6 + (-b3 + 2*b2 + 2*b1) * q^7 + (2*b3 + 2*b2 - 2*b1 - 4) * q^8 + (-2*b2 + 2*b1 + 1) * q^9 $$q - \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{6} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{7} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{8} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{9} + (\beta_{3} + \beta_1 - 2) q^{10} + ( - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{11} + ( - 4 \beta_1 + 8) q^{12} + (2 \beta_{2} - 2 \beta_1 - 6) q^{13} + (3 \beta_{3} - \beta_{2} - \beta_1 + 10) q^{14} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{15} + 8 \beta_1 q^{16} + ( - 8 \beta_{2} + 8 \beta_1 + 2) q^{17} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{18} + (4 \beta_{2} + 4 \beta_1) q^{19} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 6) q^{20} + (10 \beta_{2} - 10 \beta_1) q^{21} + ( - 2 \beta_{3} + 6 \beta_{2} + \cdots - 28) q^{22}+ \cdots + ( - 10 \beta_{3} - 12 \beta_{2} - 12 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^2 + b3 * q^3 + (-b3 + b2 - b1 - 2) * q^4 + (b2 - b1 - 1) * q^5 + (-b3 - b2 + 3*b1 + 2) * q^6 + (-b3 + 2*b2 + 2*b1) * q^7 + (2*b3 + 2*b2 - 2*b1 - 4) * q^8 + (-2*b2 + 2*b1 + 1) * q^9 + (b3 + b1 - 2) * q^10 + (-2*b3 - 4*b2 - 4*b1) * q^11 + (-4*b1 + 8) * q^12 + (2*b2 - 2*b1 - 6) * q^13 + (3*b3 - b2 - b1 + 10) * q^14 + (b3 - 2*b2 - 2*b1) * q^15 + 8*b1 * q^16 + (-8*b2 + 8*b1 + 2) * q^17 + (-2*b3 + b2 - 2*b1 + 4) * q^18 + (4*b2 + 4*b1) * q^19 + (-b3 + b2 + 3*b1 + 6) * q^20 + (10*b2 - 10*b1) * q^21 + (-2*b3 + 6*b2 - 10*b1 - 28) * q^22 + (-3*b3 + 2*b2 + 2*b1) * q^23 + (-8*b2 - 16) * q^24 + 5 * q^25 + (2*b3 + 4*b2 + 2*b1 - 4) * q^26 + (6*b3 + 4*b2 + 4*b1) * q^27 + (-4*b3 - 12*b2 + 8*b1) * q^28 + (4*b2 - 4*b1 - 6) * q^29 + (-3*b3 + b2 + b1 - 10) * q^30 + (10*b3 - 4*b2 - 4*b1) * q^31 + 32 * q^32 + (-12*b2 + 12*b1 + 32) * q^33 + (-8*b3 + 6*b2 - 8*b1 + 16) * q^34 - 5*b3 * q^35 + (3*b3 - 3*b2 - 5*b1 - 10) * q^36 + (10*b2 - 10*b1 - 6) * q^37 + (4*b3 - 4*b2 + 4*b1 + 24) * q^38 + (-2*b3 - 4*b2 - 4*b1) * q^39 + (2*b3 - 6*b2 - 2*b1 + 12) * q^40 + (-6*b2 + 6*b1 - 22) * q^41 + (10*b3 - 10*b2 + 10*b1 - 20) * q^42 + (-3*b3 - 4*b2 - 4*b1) * q^43 + (8*b3 + 24*b2 - 32) * q^44 + (-b2 + b1 - 9) * q^45 + (5*b3 + b2 - 7*b1 + 6) * q^46 + (-13*b3 - 6*b2 - 6*b1) * q^47 + (-8*b3 + 24*b2 - 8*b1 - 16) * q^48 + (-10*b2 + 10*b1 + 9) * q^49 - 5*b2 * q^50 + (-14*b3 + 16*b2 + 16*b1) * q^51 + (2*b3 - 2*b2 + 10*b1 + 20) * q^52 + (10*b2 - 10*b1 - 54) * q^53 + (-2*b3 - 10*b2 + 22*b1 + 36) * q^54 + (6*b3 + 8*b2 + 8*b1) * q^55 + (-8*b3 + 16*b2 - 24*b1) * q^56 + (16*b2 - 16*b1 - 16) * q^57 + (4*b3 + 2*b2 + 4*b1 - 8) * q^58 + (12*b3 - 12*b2 - 12*b1) * q^59 + (4*b3 + 12*b2 - 8*b1) * q^60 + (-26*b2 + 26*b1 + 58) * q^61 + (-14*b3 - 6*b2 + 26*b1 - 4) * q^62 + (11*b3 - 2*b2 - 2*b1) * q^63 - 32*b2 * q^64 + (-4*b2 + 4*b1 + 14) * q^65 + (-12*b3 - 20*b2 - 12*b1 + 24) * q^66 + (11*b3 - 20*b2 - 20*b1) * q^67 + (14*b3 - 14*b2 - 18*b1 - 36) * q^68 + (14*b2 - 14*b1 + 16) * q^69 + (5*b3 + 5*b2 - 15*b1 - 10) * q^70 + (2*b3 + 20*b2 + 20*b1) * q^71 + (-6*b3 + 10*b2 + 6*b1 - 20) * q^72 + (32*b2 - 32*b1 + 34) * q^73 + (10*b3 - 4*b2 + 10*b1 - 20) * q^74 + 5*b3 * q^75 + (-8*b3 - 24*b2 + 8*b1 + 16) * q^76 + (-20*b2 + 20*b1 + 80) * q^77 + (-2*b3 + 6*b2 - 10*b1 - 28) * q^78 + (-20*b3 + 16*b2 + 16*b1) * q^79 + (-8*b3 - 8*b2 - 16) * q^80 + (-14*b2 + 14*b1 - 55) * q^81 + (-6*b3 + 28*b2 - 6*b1 + 12) * q^82 + (13*b3 + 12*b2 + 12*b1) * q^83 + (-20*b3 + 20*b2 + 20*b1 + 40) * q^84 + (-6*b2 + 6*b1 - 34) * q^85 + (-b3 + 7*b2 - 13*b1 - 30) * q^86 + (2*b3 - 8*b2 - 8*b1) * q^87 + (16*b3 + 48*b1 + 64) * q^88 + (40*b2 - 40*b1 - 62) * q^89 + (-b3 + 10*b2 - b1 + 2) * q^90 + (-6*b3 - 8*b2 - 8*b1) * q^91 + (-4*b3 - 12*b2 + 16*b1 - 16) * q^92 + (-36*b2 + 36*b1 - 64) * q^93 + (7*b3 + 19*b2 - 45*b1 - 62) * q^94 + (-8*b3 - 4*b2 - 4*b1) * q^95 + 32*b3 * q^96 + (12*b2 - 12*b1 - 78) * q^97 + (-10*b3 + b2 - 10*b1 + 20) * q^98 + (-10*b3 - 12*b2 - 12*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 4 q^{4} - 8 q^{8} - 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 4 * q^4 - 8 * q^8 - 4 * q^9 $$4 q - 2 q^{2} - 4 q^{4} - 8 q^{8} - 4 q^{9} - 10 q^{10} + 40 q^{12} - 16 q^{13} + 40 q^{14} - 16 q^{16} - 24 q^{17} + 22 q^{18} + 20 q^{20} + 40 q^{21} - 80 q^{22} - 80 q^{24} + 20 q^{25} - 12 q^{26} - 40 q^{28} - 8 q^{29} - 40 q^{30} + 128 q^{32} + 80 q^{33} + 92 q^{34} - 36 q^{36} + 16 q^{37} + 80 q^{38} + 40 q^{40} - 112 q^{41} - 120 q^{42} - 80 q^{44} - 40 q^{45} + 40 q^{46} - 4 q^{49} - 10 q^{50} + 56 q^{52} - 176 q^{53} + 80 q^{54} + 80 q^{56} - 36 q^{58} + 40 q^{60} + 128 q^{61} - 80 q^{62} - 64 q^{64} + 40 q^{65} + 80 q^{66} - 136 q^{68} + 120 q^{69} - 72 q^{72} + 264 q^{73} - 108 q^{74} + 240 q^{77} - 80 q^{78} - 80 q^{80} - 276 q^{81} + 116 q^{82} + 160 q^{84} - 160 q^{85} - 80 q^{86} + 160 q^{88} - 88 q^{89} + 30 q^{90} - 120 q^{92} - 400 q^{93} - 120 q^{94} - 264 q^{97} + 102 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 4 * q^4 - 8 * q^8 - 4 * q^9 - 10 * q^10 + 40 * q^12 - 16 * q^13 + 40 * q^14 - 16 * q^16 - 24 * q^17 + 22 * q^18 + 20 * q^20 + 40 * q^21 - 80 * q^22 - 80 * q^24 + 20 * q^25 - 12 * q^26 - 40 * q^28 - 8 * q^29 - 40 * q^30 + 128 * q^32 + 80 * q^33 + 92 * q^34 - 36 * q^36 + 16 * q^37 + 80 * q^38 + 40 * q^40 - 112 * q^41 - 120 * q^42 - 80 * q^44 - 40 * q^45 + 40 * q^46 - 4 * q^49 - 10 * q^50 + 56 * q^52 - 176 * q^53 + 80 * q^54 + 80 * q^56 - 36 * q^58 + 40 * q^60 + 128 * q^61 - 80 * q^62 - 64 * q^64 + 40 * q^65 + 80 * q^66 - 136 * q^68 + 120 * q^69 - 72 * q^72 + 264 * q^73 - 108 * q^74 + 240 * q^77 - 80 * q^78 - 80 * q^80 - 276 * q^81 + 116 * q^82 + 160 * q^84 - 160 * q^85 - 80 * q^86 + 160 * q^88 - 88 * q^89 + 30 * q^90 - 120 * q^92 - 400 * q^93 - 120 * q^94 - 264 * q^97 + 102 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{10}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$2\zeta_{10}^{3}$$ 2*v^3 $$\beta_{3}$$ $$=$$ $$2\zeta_{10}^{3} - 2\zeta_{10}^{2} + 4\zeta_{10} - 2$$ 2*v^3 - 2*v^2 + 4*v - 2
 $$\zeta_{10}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 4$$ (b3 - b2 + b1 + 2) / 4 $$\zeta_{10}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{10}^{3}$$ $$=$$ $$( \beta_{2} ) / 2$$ (b2) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i
−1.61803 1.17557i 3.80423i 1.23607 + 3.80423i 2.23607 −4.47214 + 6.15537i 8.50651i 2.47214 7.60845i −5.47214 −3.61803 2.62866i
11.2 −1.61803 + 1.17557i 3.80423i 1.23607 3.80423i 2.23607 −4.47214 6.15537i 8.50651i 2.47214 + 7.60845i −5.47214 −3.61803 + 2.62866i
11.3 0.618034 1.90211i 2.35114i −3.23607 2.35114i −2.23607 4.47214 + 1.45309i 5.25731i −6.47214 + 4.70228i 3.47214 −1.38197 + 4.25325i
11.4 0.618034 + 1.90211i 2.35114i −3.23607 + 2.35114i −2.23607 4.47214 1.45309i 5.25731i −6.47214 4.70228i 3.47214 −1.38197 4.25325i
 $$n$$: 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.3.b.a 4
3.b odd 2 1 180.3.c.a 4
4.b odd 2 1 inner 20.3.b.a 4
5.b even 2 1 100.3.b.f 4
5.c odd 4 2 100.3.d.b 8
8.b even 2 1 320.3.b.c 4
8.d odd 2 1 320.3.b.c 4
12.b even 2 1 180.3.c.a 4
15.d odd 2 1 900.3.c.k 4
15.e even 4 2 900.3.f.e 8
16.e even 4 2 1280.3.g.e 8
16.f odd 4 2 1280.3.g.e 8
20.d odd 2 1 100.3.b.f 4
20.e even 4 2 100.3.d.b 8
24.f even 2 1 2880.3.e.e 4
24.h odd 2 1 2880.3.e.e 4
40.e odd 2 1 1600.3.b.s 4
40.f even 2 1 1600.3.b.s 4
40.i odd 4 2 1600.3.h.n 8
40.k even 4 2 1600.3.h.n 8
60.h even 2 1 900.3.c.k 4
60.l odd 4 2 900.3.f.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 1.a even 1 1 trivial
20.3.b.a 4 4.b odd 2 1 inner
100.3.b.f 4 5.b even 2 1
100.3.b.f 4 20.d odd 2 1
100.3.d.b 8 5.c odd 4 2
100.3.d.b 8 20.e even 4 2
180.3.c.a 4 3.b odd 2 1
180.3.c.a 4 12.b even 2 1
320.3.b.c 4 8.b even 2 1
320.3.b.c 4 8.d odd 2 1
900.3.c.k 4 15.d odd 2 1
900.3.c.k 4 60.h even 2 1
900.3.f.e 8 15.e even 4 2
900.3.f.e 8 60.l odd 4 2
1280.3.g.e 8 16.e even 4 2
1280.3.g.e 8 16.f odd 4 2
1600.3.b.s 4 40.e odd 2 1
1600.3.b.s 4 40.f even 2 1
1600.3.h.n 8 40.i odd 4 2
1600.3.h.n 8 40.k even 4 2
2880.3.e.e 4 24.f even 2 1
2880.3.e.e 4 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(20, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + \cdots + 16$$
$3$ $$T^{4} + 20T^{2} + 80$$
$5$ $$(T^{2} - 5)^{2}$$
$7$ $$T^{4} + 100T^{2} + 2000$$
$11$ $$T^{4} + 400T^{2} + 1280$$
$13$ $$(T^{2} + 8 T - 4)^{2}$$
$17$ $$(T^{2} + 12 T - 284)^{2}$$
$19$ $$T^{4} + 320 T^{2} + 20480$$
$23$ $$T^{4} + 260T^{2} + 80$$
$29$ $$(T^{2} + 4 T - 76)^{2}$$
$31$ $$T^{4} + 2320 T^{2} + 154880$$
$37$ $$(T^{2} - 8 T - 484)^{2}$$
$41$ $$(T^{2} + 56 T + 604)^{2}$$
$43$ $$T^{4} + 500T^{2} + 2000$$
$47$ $$T^{4} + 4100 T^{2} + 3561680$$
$53$ $$(T^{2} + 88 T + 1436)^{2}$$
$59$ $$T^{4} + 5760 T^{2} + 1658880$$
$61$ $$(T^{2} - 64 T - 2356)^{2}$$
$67$ $$T^{4} + 10420 T^{2} + 19920080$$
$71$ $$T^{4} + 8080 T^{2} + 10138880$$
$73$ $$(T^{2} - 132 T - 764)^{2}$$
$79$ $$T^{4} + 13120 T^{2} + 2478080$$
$83$ $$T^{4} + 6260 T^{2} + 2620880$$
$89$ $$(T^{2} + 44 T - 7516)^{2}$$
$97$ $$(T^{2} + 132 T + 3636)^{2}$$