Properties

 Label 70.2.e.c Level $70$ Weight $2$ Character orbit 70.e Analytic conductor $0.559$ 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] = [70,2,Mod(11,70)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(70, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("70.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$70 = 2 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 70.e (of order $$3$$, 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: $$0.558952814149$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} - q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + z * q^2 + (z - 1) * q^3 + (z - 1) * q^4 + z * q^5 - q^6 + (-3*z + 1) * q^7 - q^8 + 2*z * q^9 $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} - q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8} + 2 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + ( - 6 \zeta_{6} + 6) q^{11} - \zeta_{6} q^{12} - 4 q^{13} + ( - 2 \zeta_{6} + 3) q^{14} - q^{15} - \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{18} - 2 \zeta_{6} q^{19} - q^{20} + (\zeta_{6} + 2) q^{21} + 6 q^{22} + 3 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} + (\zeta_{6} - 1) q^{25} - 4 \zeta_{6} q^{26} - 5 q^{27} + (\zeta_{6} + 2) q^{28} - 3 q^{29} - \zeta_{6} q^{30} + (8 \zeta_{6} - 8) q^{31} + ( - \zeta_{6} + 1) q^{32} + 6 \zeta_{6} q^{33} + ( - 2 \zeta_{6} + 3) q^{35} - 2 q^{36} + 4 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{38} + ( - 4 \zeta_{6} + 4) q^{39} - \zeta_{6} q^{40} + 9 q^{41} + (3 \zeta_{6} - 1) q^{42} - 7 q^{43} + 6 \zeta_{6} q^{44} + (2 \zeta_{6} - 2) q^{45} + (3 \zeta_{6} - 3) q^{46} + q^{48} + (3 \zeta_{6} - 8) q^{49} - q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} - 5 \zeta_{6} q^{54} + 6 q^{55} + (3 \zeta_{6} - 1) q^{56} + 2 q^{57} - 3 \zeta_{6} q^{58} + ( - 6 \zeta_{6} + 6) q^{59} + ( - \zeta_{6} + 1) q^{60} - 5 \zeta_{6} q^{61} - 8 q^{62} + ( - 4 \zeta_{6} + 6) q^{63} + q^{64} - 4 \zeta_{6} q^{65} + (6 \zeta_{6} - 6) q^{66} + (5 \zeta_{6} - 5) q^{67} - 3 q^{69} + (\zeta_{6} + 2) q^{70} - 6 q^{71} - 2 \zeta_{6} q^{72} + ( - 16 \zeta_{6} + 16) q^{73} + (4 \zeta_{6} - 4) q^{74} - \zeta_{6} q^{75} + 2 q^{76} + ( - 6 \zeta_{6} - 12) q^{77} + 4 q^{78} - 2 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} + 9 \zeta_{6} q^{82} + 3 q^{83} + (2 \zeta_{6} - 3) q^{84} - 7 \zeta_{6} q^{86} + ( - 3 \zeta_{6} + 3) q^{87} + (6 \zeta_{6} - 6) q^{88} + 15 \zeta_{6} q^{89} - 2 q^{90} + (12 \zeta_{6} - 4) q^{91} - 3 q^{92} - 8 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{95} + \zeta_{6} q^{96} + 14 q^{97} + ( - 5 \zeta_{6} - 3) q^{98} + 12 q^{99} +O(q^{100})$$ q + z * q^2 + (z - 1) * q^3 + (z - 1) * q^4 + z * q^5 - q^6 + (-3*z + 1) * q^7 - q^8 + 2*z * q^9 + (z - 1) * q^10 + (-6*z + 6) * q^11 - z * q^12 - 4 * q^13 + (-2*z + 3) * q^14 - q^15 - z * q^16 + (2*z - 2) * q^18 - 2*z * q^19 - q^20 + (z + 2) * q^21 + 6 * q^22 + 3*z * q^23 + (-z + 1) * q^24 + (z - 1) * q^25 - 4*z * q^26 - 5 * q^27 + (z + 2) * q^28 - 3 * q^29 - z * q^30 + (8*z - 8) * q^31 + (-z + 1) * q^32 + 6*z * q^33 + (-2*z + 3) * q^35 - 2 * q^36 + 4*z * q^37 + (-2*z + 2) * q^38 + (-4*z + 4) * q^39 - z * q^40 + 9 * q^41 + (3*z - 1) * q^42 - 7 * q^43 + 6*z * q^44 + (2*z - 2) * q^45 + (3*z - 3) * q^46 + q^48 + (3*z - 8) * q^49 - q^50 + (-4*z + 4) * q^52 + (-6*z + 6) * q^53 - 5*z * q^54 + 6 * q^55 + (3*z - 1) * q^56 + 2 * q^57 - 3*z * q^58 + (-6*z + 6) * q^59 + (-z + 1) * q^60 - 5*z * q^61 - 8 * q^62 + (-4*z + 6) * q^63 + q^64 - 4*z * q^65 + (6*z - 6) * q^66 + (5*z - 5) * q^67 - 3 * q^69 + (z + 2) * q^70 - 6 * q^71 - 2*z * q^72 + (-16*z + 16) * q^73 + (4*z - 4) * q^74 - z * q^75 + 2 * q^76 + (-6*z - 12) * q^77 + 4 * q^78 - 2*z * q^79 + (-z + 1) * q^80 + (z - 1) * q^81 + 9*z * q^82 + 3 * q^83 + (2*z - 3) * q^84 - 7*z * q^86 + (-3*z + 3) * q^87 + (6*z - 6) * q^88 + 15*z * q^89 - 2 * q^90 + (12*z - 4) * q^91 - 3 * q^92 - 8*z * q^93 + (-2*z + 2) * q^95 + z * q^96 + 14 * q^97 + (-5*z - 3) * q^98 + 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 - q^3 - q^4 + q^5 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 $$2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + 6 q^{11} - q^{12} - 8 q^{13} + 4 q^{14} - 2 q^{15} - q^{16} - 2 q^{18} - 2 q^{19} - 2 q^{20} + 5 q^{21} + 12 q^{22} + 3 q^{23} + q^{24} - q^{25} - 4 q^{26} - 10 q^{27} + 5 q^{28} - 6 q^{29} - q^{30} - 8 q^{31} + q^{32} + 6 q^{33} + 4 q^{35} - 4 q^{36} + 4 q^{37} + 2 q^{38} + 4 q^{39} - q^{40} + 18 q^{41} + q^{42} - 14 q^{43} + 6 q^{44} - 2 q^{45} - 3 q^{46} + 2 q^{48} - 13 q^{49} - 2 q^{50} + 4 q^{52} + 6 q^{53} - 5 q^{54} + 12 q^{55} + q^{56} + 4 q^{57} - 3 q^{58} + 6 q^{59} + q^{60} - 5 q^{61} - 16 q^{62} + 8 q^{63} + 2 q^{64} - 4 q^{65} - 6 q^{66} - 5 q^{67} - 6 q^{69} + 5 q^{70} - 12 q^{71} - 2 q^{72} + 16 q^{73} - 4 q^{74} - q^{75} + 4 q^{76} - 30 q^{77} + 8 q^{78} - 2 q^{79} + q^{80} - q^{81} + 9 q^{82} + 6 q^{83} - 4 q^{84} - 7 q^{86} + 3 q^{87} - 6 q^{88} + 15 q^{89} - 4 q^{90} + 4 q^{91} - 6 q^{92} - 8 q^{93} + 2 q^{95} + q^{96} + 28 q^{97} - 11 q^{98} + 24 q^{99}+O(q^{100})$$ 2 * q + q^2 - q^3 - q^4 + q^5 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 - q^10 + 6 * q^11 - q^12 - 8 * q^13 + 4 * q^14 - 2 * q^15 - q^16 - 2 * q^18 - 2 * q^19 - 2 * q^20 + 5 * q^21 + 12 * q^22 + 3 * q^23 + q^24 - q^25 - 4 * q^26 - 10 * q^27 + 5 * q^28 - 6 * q^29 - q^30 - 8 * q^31 + q^32 + 6 * q^33 + 4 * q^35 - 4 * q^36 + 4 * q^37 + 2 * q^38 + 4 * q^39 - q^40 + 18 * q^41 + q^42 - 14 * q^43 + 6 * q^44 - 2 * q^45 - 3 * q^46 + 2 * q^48 - 13 * q^49 - 2 * q^50 + 4 * q^52 + 6 * q^53 - 5 * q^54 + 12 * q^55 + q^56 + 4 * q^57 - 3 * q^58 + 6 * q^59 + q^60 - 5 * q^61 - 16 * q^62 + 8 * q^63 + 2 * q^64 - 4 * q^65 - 6 * q^66 - 5 * q^67 - 6 * q^69 + 5 * q^70 - 12 * q^71 - 2 * q^72 + 16 * q^73 - 4 * q^74 - q^75 + 4 * q^76 - 30 * q^77 + 8 * q^78 - 2 * q^79 + q^80 - q^81 + 9 * q^82 + 6 * q^83 - 4 * q^84 - 7 * q^86 + 3 * q^87 - 6 * q^88 + 15 * q^89 - 4 * q^90 + 4 * q^91 - 6 * q^92 - 8 * q^93 + 2 * q^95 + q^96 + 28 * q^97 - 11 * q^98 + 24 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/70\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$57$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

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
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 −0.500000 2.59808i −1.00000 1.00000 + 1.73205i −0.500000 + 0.866025i
51.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.00000 −0.500000 + 2.59808i −1.00000 1.00000 1.73205i −0.500000 0.866025i
 $$n$$: 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
7.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.e.c 2
3.b odd 2 1 630.2.k.b 2
4.b odd 2 1 560.2.q.g 2
5.b even 2 1 350.2.e.e 2
5.c odd 4 2 350.2.j.b 4
7.b odd 2 1 490.2.e.h 2
7.c even 3 1 inner 70.2.e.c 2
7.c even 3 1 490.2.a.c 1
7.d odd 6 1 490.2.a.b 1
7.d odd 6 1 490.2.e.h 2
21.g even 6 1 4410.2.a.bd 1
21.h odd 6 1 630.2.k.b 2
21.h odd 6 1 4410.2.a.bm 1
28.f even 6 1 3920.2.a.bc 1
28.g odd 6 1 560.2.q.g 2
28.g odd 6 1 3920.2.a.p 1
35.i odd 6 1 2450.2.a.bc 1
35.j even 6 1 350.2.e.e 2
35.j even 6 1 2450.2.a.w 1
35.k even 12 2 2450.2.c.l 2
35.l odd 12 2 350.2.j.b 4
35.l odd 12 2 2450.2.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 1.a even 1 1 trivial
70.2.e.c 2 7.c even 3 1 inner
350.2.e.e 2 5.b even 2 1
350.2.e.e 2 35.j even 6 1
350.2.j.b 4 5.c odd 4 2
350.2.j.b 4 35.l odd 12 2
490.2.a.b 1 7.d odd 6 1
490.2.a.c 1 7.c even 3 1
490.2.e.h 2 7.b odd 2 1
490.2.e.h 2 7.d odd 6 1
560.2.q.g 2 4.b odd 2 1
560.2.q.g 2 28.g odd 6 1
630.2.k.b 2 3.b odd 2 1
630.2.k.b 2 21.h odd 6 1
2450.2.a.w 1 35.j even 6 1
2450.2.a.bc 1 35.i odd 6 1
2450.2.c.g 2 35.l odd 12 2
2450.2.c.l 2 35.k even 12 2
3920.2.a.p 1 28.g odd 6 1
3920.2.a.bc 1 28.f even 6 1
4410.2.a.bd 1 21.g even 6 1
4410.2.a.bm 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(70, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{13} + 4$$ T13 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$(T + 3)^{2}$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$T^{2} - 4T + 16$$
$41$ $$(T - 9)^{2}$$
$43$ $$(T + 7)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} - 16T + 256$$
$79$ $$T^{2} + 2T + 4$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} - 15T + 225$$
$97$ $$(T - 14)^{2}$$