# Properties

 Label 980.3.l.a Level $980$ Weight $3$ Character orbit 980.l Analytic conductor $26.703$ 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] = [980,3,Mod(197,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 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("980.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 980.l (of order $$4$$, degree $$2$$, 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: $$26.7030659073$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i - 1) q^{3} + ( - 4 i + 3) q^{5} + 7 i q^{9}+O(q^{10})$$ q + (i - 1) * q^3 + (-4*i + 3) * q^5 + 7*i * q^9 $$q + (i - 1) q^{3} + ( - 4 i + 3) q^{5} + 7 i q^{9} + 10 q^{11} + (9 i - 9) q^{13} + (7 i + 1) q^{15} + ( - i - 1) q^{17} + 8 i q^{19} + (23 i - 23) q^{23} + ( - 24 i - 7) q^{25} + ( - 16 i - 16) q^{27} - 8 i q^{29} + 14 q^{31} + (10 i - 10) q^{33} + (33 i + 33) q^{37} - 18 i q^{39} + 14 q^{41} + (15 i - 15) q^{43} + (21 i + 28) q^{45} + (39 i + 39) q^{47} + 2 q^{51} + (7 i - 7) q^{53} + ( - 40 i + 30) q^{55} + ( - 8 i - 8) q^{57} + 56 i q^{59} - 42 q^{61} + (63 i + 9) q^{65} + ( - 7 i - 7) q^{67} - 46 i q^{69} + 98 q^{71} + (49 i - 49) q^{73} + (17 i + 31) q^{75} + 96 i q^{79} - 31 q^{81} + ( - 63 i + 63) q^{83} + (i - 7) q^{85} + (8 i + 8) q^{87} + 112 i q^{89} + (14 i - 14) q^{93} + (24 i + 32) q^{95} + ( - 33 i - 33) q^{97} + 70 i q^{99} +O(q^{100})$$ q + (i - 1) * q^3 + (-4*i + 3) * q^5 + 7*i * q^9 + 10 * q^11 + (9*i - 9) * q^13 + (7*i + 1) * q^15 + (-i - 1) * q^17 + 8*i * q^19 + (23*i - 23) * q^23 + (-24*i - 7) * q^25 + (-16*i - 16) * q^27 - 8*i * q^29 + 14 * q^31 + (10*i - 10) * q^33 + (33*i + 33) * q^37 - 18*i * q^39 + 14 * q^41 + (15*i - 15) * q^43 + (21*i + 28) * q^45 + (39*i + 39) * q^47 + 2 * q^51 + (7*i - 7) * q^53 + (-40*i + 30) * q^55 + (-8*i - 8) * q^57 + 56*i * q^59 - 42 * q^61 + (63*i + 9) * q^65 + (-7*i - 7) * q^67 - 46*i * q^69 + 98 * q^71 + (49*i - 49) * q^73 + (17*i + 31) * q^75 + 96*i * q^79 - 31 * q^81 + (-63*i + 63) * q^83 + (i - 7) * q^85 + (8*i + 8) * q^87 + 112*i * q^89 + (14*i - 14) * q^93 + (24*i + 32) * q^95 + (-33*i - 33) * q^97 + 70*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 6 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 + 6 * q^5 $$2 q - 2 q^{3} + 6 q^{5} + 20 q^{11} - 18 q^{13} + 2 q^{15} - 2 q^{17} - 46 q^{23} - 14 q^{25} - 32 q^{27} + 28 q^{31} - 20 q^{33} + 66 q^{37} + 28 q^{41} - 30 q^{43} + 56 q^{45} + 78 q^{47} + 4 q^{51} - 14 q^{53} + 60 q^{55} - 16 q^{57} - 84 q^{61} + 18 q^{65} - 14 q^{67} + 196 q^{71} - 98 q^{73} + 62 q^{75} - 62 q^{81} + 126 q^{83} - 14 q^{85} + 16 q^{87} - 28 q^{93} + 64 q^{95} - 66 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 6 * q^5 + 20 * q^11 - 18 * q^13 + 2 * q^15 - 2 * q^17 - 46 * q^23 - 14 * q^25 - 32 * q^27 + 28 * q^31 - 20 * q^33 + 66 * q^37 + 28 * q^41 - 30 * q^43 + 56 * q^45 + 78 * q^47 + 4 * q^51 - 14 * q^53 + 60 * q^55 - 16 * q^57 - 84 * q^61 + 18 * q^65 - 14 * q^67 + 196 * q^71 - 98 * q^73 + 62 * q^75 - 62 * q^81 + 126 * q^83 - 14 * q^85 + 16 * q^87 - 28 * q^93 + 64 * q^95 - 66 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$1$$ $$i$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 1.00000i − 1.00000i
0 −1.00000 + 1.00000i 0 3.00000 4.00000i 0 0 0 7.00000i 0
393.1 0 −1.00000 1.00000i 0 3.00000 + 4.00000i 0 0 0 7.00000i 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.3.l.a 2
5.c odd 4 1 inner 980.3.l.a 2
7.b odd 2 1 20.3.f.a 2
21.c even 2 1 180.3.l.a 2
28.d even 2 1 80.3.p.a 2
35.c odd 2 1 100.3.f.a 2
35.f even 4 1 20.3.f.a 2
35.f even 4 1 100.3.f.a 2
56.e even 2 1 320.3.p.g 2
56.h odd 2 1 320.3.p.c 2
84.h odd 2 1 720.3.bh.e 2
105.g even 2 1 900.3.l.a 2
105.k odd 4 1 180.3.l.a 2
105.k odd 4 1 900.3.l.a 2
140.c even 2 1 400.3.p.d 2
140.j odd 4 1 80.3.p.a 2
140.j odd 4 1 400.3.p.d 2
280.s even 4 1 320.3.p.c 2
280.y odd 4 1 320.3.p.g 2
420.w even 4 1 720.3.bh.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.f.a 2 7.b odd 2 1
20.3.f.a 2 35.f even 4 1
80.3.p.a 2 28.d even 2 1
80.3.p.a 2 140.j odd 4 1
100.3.f.a 2 35.c odd 2 1
100.3.f.a 2 35.f even 4 1
180.3.l.a 2 21.c even 2 1
180.3.l.a 2 105.k odd 4 1
320.3.p.c 2 56.h odd 2 1
320.3.p.c 2 280.s even 4 1
320.3.p.g 2 56.e even 2 1
320.3.p.g 2 280.y odd 4 1
400.3.p.d 2 140.c even 2 1
400.3.p.d 2 140.j odd 4 1
720.3.bh.e 2 84.h odd 2 1
720.3.bh.e 2 420.w even 4 1
900.3.l.a 2 105.g even 2 1
900.3.l.a 2 105.k odd 4 1
980.3.l.a 2 1.a even 1 1 trivial
980.3.l.a 2 5.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 2$$ acting on $$S_{3}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2} - 6T + 25$$
$7$ $$T^{2}$$
$11$ $$(T - 10)^{2}$$
$13$ $$T^{2} + 18T + 162$$
$17$ $$T^{2} + 2T + 2$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2} + 46T + 1058$$
$29$ $$T^{2} + 64$$
$31$ $$(T - 14)^{2}$$
$37$ $$T^{2} - 66T + 2178$$
$41$ $$(T - 14)^{2}$$
$43$ $$T^{2} + 30T + 450$$
$47$ $$T^{2} - 78T + 3042$$
$53$ $$T^{2} + 14T + 98$$
$59$ $$T^{2} + 3136$$
$61$ $$(T + 42)^{2}$$
$67$ $$T^{2} + 14T + 98$$
$71$ $$(T - 98)^{2}$$
$73$ $$T^{2} + 98T + 4802$$
$79$ $$T^{2} + 9216$$
$83$ $$T^{2} - 126T + 7938$$
$89$ $$T^{2} + 12544$$
$97$ $$T^{2} + 66T + 2178$$