Properties

 Label 28.2.f.a Level $28$ Weight $2$ Character orbit 28.f Analytic conductor $0.224$ Analytic rank $0$ Dimension $4$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,2,Mod(3,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("28.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 28.f (of order $$6$$, 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: $$0.223581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/28\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{2}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−1.36603 0.366025i 0.866025 1.50000i 1.73205 + 1.00000i −1.50000 + 0.866025i −1.73205 + 1.73205i −1.73205 + 2.00000i −2.00000 2.00000i 0 2.36603 0.633975i
3.2 0.366025 1.36603i −0.866025 + 1.50000i −1.73205 1.00000i −1.50000 + 0.866025i 1.73205 + 1.73205i 1.73205 2.00000i −2.00000 + 2.00000i 0 0.633975 + 2.36603i
19.1 −1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 1.00000i −1.50000 0.866025i −1.73205 1.73205i −1.73205 2.00000i −2.00000 + 2.00000i 0 2.36603 + 0.633975i
19.2 0.366025 + 1.36603i −0.866025 1.50000i −1.73205 + 1.00000i −1.50000 0.866025i 1.73205 1.73205i 1.73205 + 2.00000i −2.00000 2.00000i 0 0.633975 2.36603i
 $$n$$: 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
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.2.f.a 4
3.b odd 2 1 252.2.bf.e 4
4.b odd 2 1 inner 28.2.f.a 4
5.b even 2 1 700.2.p.a 4
5.c odd 4 1 700.2.t.a 4
5.c odd 4 1 700.2.t.b 4
7.b odd 2 1 196.2.f.a 4
7.c even 3 1 196.2.d.b 4
7.c even 3 1 196.2.f.a 4
7.d odd 6 1 inner 28.2.f.a 4
7.d odd 6 1 196.2.d.b 4
8.b even 2 1 448.2.p.d 4
8.d odd 2 1 448.2.p.d 4
12.b even 2 1 252.2.bf.e 4
20.d odd 2 1 700.2.p.a 4
20.e even 4 1 700.2.t.a 4
20.e even 4 1 700.2.t.b 4
21.g even 6 1 252.2.bf.e 4
21.g even 6 1 1764.2.b.a 4
21.h odd 6 1 1764.2.b.a 4
28.d even 2 1 196.2.f.a 4
28.f even 6 1 inner 28.2.f.a 4
28.f even 6 1 196.2.d.b 4
28.g odd 6 1 196.2.d.b 4
28.g odd 6 1 196.2.f.a 4
35.i odd 6 1 700.2.p.a 4
35.k even 12 1 700.2.t.a 4
35.k even 12 1 700.2.t.b 4
56.j odd 6 1 448.2.p.d 4
56.j odd 6 1 3136.2.f.e 4
56.k odd 6 1 3136.2.f.e 4
56.m even 6 1 448.2.p.d 4
56.m even 6 1 3136.2.f.e 4
56.p even 6 1 3136.2.f.e 4
84.j odd 6 1 252.2.bf.e 4
84.j odd 6 1 1764.2.b.a 4
84.n even 6 1 1764.2.b.a 4
140.s even 6 1 700.2.p.a 4
140.x odd 12 1 700.2.t.a 4
140.x odd 12 1 700.2.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 1.a even 1 1 trivial
28.2.f.a 4 4.b odd 2 1 inner
28.2.f.a 4 7.d odd 6 1 inner
28.2.f.a 4 28.f even 6 1 inner
196.2.d.b 4 7.c even 3 1
196.2.d.b 4 7.d odd 6 1
196.2.d.b 4 28.f even 6 1
196.2.d.b 4 28.g odd 6 1
196.2.f.a 4 7.b odd 2 1
196.2.f.a 4 7.c even 3 1
196.2.f.a 4 28.d even 2 1
196.2.f.a 4 28.g odd 6 1
252.2.bf.e 4 3.b odd 2 1
252.2.bf.e 4 12.b even 2 1
252.2.bf.e 4 21.g even 6 1
252.2.bf.e 4 84.j odd 6 1
448.2.p.d 4 8.b even 2 1
448.2.p.d 4 8.d odd 2 1
448.2.p.d 4 56.j odd 6 1
448.2.p.d 4 56.m even 6 1
700.2.p.a 4 5.b even 2 1
700.2.p.a 4 20.d odd 2 1
700.2.p.a 4 35.i odd 6 1
700.2.p.a 4 140.s even 6 1
700.2.t.a 4 5.c odd 4 1
700.2.t.a 4 20.e even 4 1
700.2.t.a 4 35.k even 12 1
700.2.t.a 4 140.x odd 12 1
700.2.t.b 4 5.c odd 4 1
700.2.t.b 4 20.e even 4 1
700.2.t.b 4 35.k even 12 1
700.2.t.b 4 140.x odd 12 1
1764.2.b.a 4 21.g even 6 1
1764.2.b.a 4 21.h odd 6 1
1764.2.b.a 4 84.j odd 6 1
1764.2.b.a 4 84.n even 6 1
3136.2.f.e 4 56.j odd 6 1
3136.2.f.e 4 56.k odd 6 1
3136.2.f.e 4 56.m even 6 1
3136.2.f.e 4 56.p even 6 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(28, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + \cdots + 4$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$(T^{2} + 3 T + 3)^{2}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$T^{4} - T^{2} + 1$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} + 3 T + 3)^{2}$$
$19$ $$T^{4} + 27T^{2} + 729$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T - 4)^{4}$$
$31$ $$T^{4} + 3T^{2} + 9$$
$37$ $$(T^{2} + 3 T + 9)^{2}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$T^{4} + 75T^{2} + 5625$$
$53$ $$(T^{2} - T + 1)^{2}$$
$59$ $$T^{4} + 27T^{2} + 729$$
$61$ $$(T^{2} + 9 T + 27)^{2}$$
$67$ $$T^{4} - 9T^{2} + 81$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$(T^{2} - 15 T + 75)^{2}$$
$79$ $$T^{4} - 81T^{2} + 6561$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$(T^{2} - 27 T + 243)^{2}$$
$97$ $$(T^{2} + 300)^{2}$$