# Properties

 Label 70.2.e.a 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + 3 q^{6} + (3 \zeta_{6} - 1) q^{7} + q^{8} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (3*z - 3) * q^3 + (z - 1) * q^4 + z * q^5 + 3 * q^6 + (3*z - 1) * q^7 + q^8 - 6*z * q^9 $$q - \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + 3 q^{6} + (3 \zeta_{6} - 1) q^{7} + q^{8} - 6 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + ( - 2 \zeta_{6} + 2) q^{11} - 3 \zeta_{6} q^{12} + ( - 2 \zeta_{6} + 3) q^{14} - 3 q^{15} - \zeta_{6} q^{16} + ( - 4 \zeta_{6} + 4) q^{17} + (6 \zeta_{6} - 6) q^{18} + 6 \zeta_{6} q^{19} - q^{20} + ( - 3 \zeta_{6} - 6) q^{21} - 2 q^{22} - 3 \zeta_{6} q^{23} + (3 \zeta_{6} - 3) q^{24} + (\zeta_{6} - 1) q^{25} + 9 q^{27} + ( - \zeta_{6} - 2) q^{28} + 9 q^{29} + 3 \zeta_{6} q^{30} + ( - 4 \zeta_{6} + 4) q^{31} + (\zeta_{6} - 1) q^{32} + 6 \zeta_{6} q^{33} - 4 q^{34} + (2 \zeta_{6} - 3) q^{35} + 6 q^{36} + 4 \zeta_{6} q^{37} + ( - 6 \zeta_{6} + 6) q^{38} + \zeta_{6} q^{40} - 7 q^{41} + (9 \zeta_{6} - 3) q^{42} - 5 q^{43} + 2 \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{45} + (3 \zeta_{6} - 3) q^{46} - 8 \zeta_{6} q^{47} + 3 q^{48} + (3 \zeta_{6} - 8) q^{49} + q^{50} + 12 \zeta_{6} q^{51} + ( - 2 \zeta_{6} + 2) q^{53} - 9 \zeta_{6} q^{54} + 2 q^{55} + (3 \zeta_{6} - 1) q^{56} - 18 q^{57} - 9 \zeta_{6} q^{58} + (10 \zeta_{6} - 10) q^{59} + ( - 3 \zeta_{6} + 3) q^{60} - \zeta_{6} q^{61} - 4 q^{62} + ( - 12 \zeta_{6} + 18) q^{63} + q^{64} + ( - 6 \zeta_{6} + 6) q^{66} + ( - 9 \zeta_{6} + 9) q^{67} + 4 \zeta_{6} q^{68} + 9 q^{69} + (\zeta_{6} + 2) q^{70} + 2 q^{71} - 6 \zeta_{6} q^{72} + ( - 4 \zeta_{6} + 4) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} - 3 \zeta_{6} q^{75} - 6 q^{76} + (2 \zeta_{6} + 4) q^{77} - 10 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} + (9 \zeta_{6} - 9) q^{81} + 7 \zeta_{6} q^{82} - 7 q^{83} + ( - 6 \zeta_{6} + 9) q^{84} + 4 q^{85} + 5 \zeta_{6} q^{86} + (27 \zeta_{6} - 27) q^{87} + ( - 2 \zeta_{6} + 2) q^{88} - \zeta_{6} q^{89} - 6 q^{90} + 3 q^{92} + 12 \zeta_{6} q^{93} + (8 \zeta_{6} - 8) q^{94} + (6 \zeta_{6} - 6) q^{95} - 3 \zeta_{6} q^{96} + 14 q^{97} + (5 \zeta_{6} + 3) q^{98} - 12 q^{99} +O(q^{100})$$ q - z * q^2 + (3*z - 3) * q^3 + (z - 1) * q^4 + z * q^5 + 3 * q^6 + (3*z - 1) * q^7 + q^8 - 6*z * q^9 + (-z + 1) * q^10 + (-2*z + 2) * q^11 - 3*z * q^12 + (-2*z + 3) * q^14 - 3 * q^15 - z * q^16 + (-4*z + 4) * q^17 + (6*z - 6) * q^18 + 6*z * q^19 - q^20 + (-3*z - 6) * q^21 - 2 * q^22 - 3*z * q^23 + (3*z - 3) * q^24 + (z - 1) * q^25 + 9 * q^27 + (-z - 2) * q^28 + 9 * q^29 + 3*z * q^30 + (-4*z + 4) * q^31 + (z - 1) * q^32 + 6*z * q^33 - 4 * q^34 + (2*z - 3) * q^35 + 6 * q^36 + 4*z * q^37 + (-6*z + 6) * q^38 + z * q^40 - 7 * q^41 + (9*z - 3) * q^42 - 5 * q^43 + 2*z * q^44 + (-6*z + 6) * q^45 + (3*z - 3) * q^46 - 8*z * q^47 + 3 * q^48 + (3*z - 8) * q^49 + q^50 + 12*z * q^51 + (-2*z + 2) * q^53 - 9*z * q^54 + 2 * q^55 + (3*z - 1) * q^56 - 18 * q^57 - 9*z * q^58 + (10*z - 10) * q^59 + (-3*z + 3) * q^60 - z * q^61 - 4 * q^62 + (-12*z + 18) * q^63 + q^64 + (-6*z + 6) * q^66 + (-9*z + 9) * q^67 + 4*z * q^68 + 9 * q^69 + (z + 2) * q^70 + 2 * q^71 - 6*z * q^72 + (-4*z + 4) * q^73 + (-4*z + 4) * q^74 - 3*z * q^75 - 6 * q^76 + (2*z + 4) * q^77 - 10*z * q^79 + (-z + 1) * q^80 + (9*z - 9) * q^81 + 7*z * q^82 - 7 * q^83 + (-6*z + 9) * q^84 + 4 * q^85 + 5*z * q^86 + (27*z - 27) * q^87 + (-2*z + 2) * q^88 - z * q^89 - 6 * q^90 + 3 * q^92 + 12*z * q^93 + (8*z - 8) * q^94 + (6*z - 6) * q^95 - 3*z * q^96 + 14 * q^97 + (5*z + 3) * q^98 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{3} - q^{4} + q^{5} + 6 q^{6} + q^{7} + 2 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q - q^2 - 3 * q^3 - q^4 + q^5 + 6 * q^6 + q^7 + 2 * q^8 - 6 * q^9 $$2 q - q^{2} - 3 q^{3} - q^{4} + q^{5} + 6 q^{6} + q^{7} + 2 q^{8} - 6 q^{9} + q^{10} + 2 q^{11} - 3 q^{12} + 4 q^{14} - 6 q^{15} - q^{16} + 4 q^{17} - 6 q^{18} + 6 q^{19} - 2 q^{20} - 15 q^{21} - 4 q^{22} - 3 q^{23} - 3 q^{24} - q^{25} + 18 q^{27} - 5 q^{28} + 18 q^{29} + 3 q^{30} + 4 q^{31} - q^{32} + 6 q^{33} - 8 q^{34} - 4 q^{35} + 12 q^{36} + 4 q^{37} + 6 q^{38} + q^{40} - 14 q^{41} + 3 q^{42} - 10 q^{43} + 2 q^{44} + 6 q^{45} - 3 q^{46} - 8 q^{47} + 6 q^{48} - 13 q^{49} + 2 q^{50} + 12 q^{51} + 2 q^{53} - 9 q^{54} + 4 q^{55} + q^{56} - 36 q^{57} - 9 q^{58} - 10 q^{59} + 3 q^{60} - q^{61} - 8 q^{62} + 24 q^{63} + 2 q^{64} + 6 q^{66} + 9 q^{67} + 4 q^{68} + 18 q^{69} + 5 q^{70} + 4 q^{71} - 6 q^{72} + 4 q^{73} + 4 q^{74} - 3 q^{75} - 12 q^{76} + 10 q^{77} - 10 q^{79} + q^{80} - 9 q^{81} + 7 q^{82} - 14 q^{83} + 12 q^{84} + 8 q^{85} + 5 q^{86} - 27 q^{87} + 2 q^{88} - q^{89} - 12 q^{90} + 6 q^{92} + 12 q^{93} - 8 q^{94} - 6 q^{95} - 3 q^{96} + 28 q^{97} + 11 q^{98} - 24 q^{99}+O(q^{100})$$ 2 * q - q^2 - 3 * q^3 - q^4 + q^5 + 6 * q^6 + q^7 + 2 * q^8 - 6 * q^9 + q^10 + 2 * q^11 - 3 * q^12 + 4 * q^14 - 6 * q^15 - q^16 + 4 * q^17 - 6 * q^18 + 6 * q^19 - 2 * q^20 - 15 * q^21 - 4 * q^22 - 3 * q^23 - 3 * q^24 - q^25 + 18 * q^27 - 5 * q^28 + 18 * q^29 + 3 * q^30 + 4 * q^31 - q^32 + 6 * q^33 - 8 * q^34 - 4 * q^35 + 12 * q^36 + 4 * q^37 + 6 * q^38 + q^40 - 14 * q^41 + 3 * q^42 - 10 * q^43 + 2 * q^44 + 6 * q^45 - 3 * q^46 - 8 * q^47 + 6 * q^48 - 13 * q^49 + 2 * q^50 + 12 * q^51 + 2 * q^53 - 9 * q^54 + 4 * q^55 + q^56 - 36 * q^57 - 9 * q^58 - 10 * q^59 + 3 * q^60 - q^61 - 8 * q^62 + 24 * q^63 + 2 * q^64 + 6 * q^66 + 9 * q^67 + 4 * q^68 + 18 * q^69 + 5 * q^70 + 4 * q^71 - 6 * q^72 + 4 * q^73 + 4 * q^74 - 3 * q^75 - 12 * q^76 + 10 * q^77 - 10 * q^79 + q^80 - 9 * q^81 + 7 * q^82 - 14 * q^83 + 12 * q^84 + 8 * q^85 + 5 * q^86 - 27 * q^87 + 2 * q^88 - q^89 - 12 * q^90 + 6 * q^92 + 12 * q^93 - 8 * q^94 - 6 * q^95 - 3 * 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 −1.50000 + 2.59808i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.00000 0.500000 + 2.59808i 1.00000 −3.00000 5.19615i 0.500000 0.866025i
51.1 −0.500000 + 0.866025i −1.50000 2.59808i −0.500000 0.866025i 0.500000 0.866025i 3.00000 0.500000 2.59808i 1.00000 −3.00000 + 5.19615i 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.a 2
3.b odd 2 1 630.2.k.f 2
4.b odd 2 1 560.2.q.i 2
5.b even 2 1 350.2.e.l 2
5.c odd 4 2 350.2.j.f 4
7.b odd 2 1 490.2.e.f 2
7.c even 3 1 inner 70.2.e.a 2
7.c even 3 1 490.2.a.k 1
7.d odd 6 1 490.2.a.e 1
7.d odd 6 1 490.2.e.f 2
21.g even 6 1 4410.2.a.h 1
21.h odd 6 1 630.2.k.f 2
21.h odd 6 1 4410.2.a.r 1
28.f even 6 1 3920.2.a.bk 1
28.g odd 6 1 560.2.q.i 2
28.g odd 6 1 3920.2.a.b 1
35.i odd 6 1 2450.2.a.q 1
35.j even 6 1 350.2.e.l 2
35.j even 6 1 2450.2.a.b 1
35.k even 12 2 2450.2.c.a 2
35.l odd 12 2 350.2.j.f 4
35.l odd 12 2 2450.2.c.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 1.a even 1 1 trivial
70.2.e.a 2 7.c even 3 1 inner
350.2.e.l 2 5.b even 2 1
350.2.e.l 2 35.j even 6 1
350.2.j.f 4 5.c odd 4 2
350.2.j.f 4 35.l odd 12 2
490.2.a.e 1 7.d odd 6 1
490.2.a.k 1 7.c even 3 1
490.2.e.f 2 7.b odd 2 1
490.2.e.f 2 7.d odd 6 1
560.2.q.i 2 4.b odd 2 1
560.2.q.i 2 28.g odd 6 1
630.2.k.f 2 3.b odd 2 1
630.2.k.f 2 21.h odd 6 1
2450.2.a.b 1 35.j even 6 1
2450.2.a.q 1 35.i odd 6 1
2450.2.c.a 2 35.k even 12 2
2450.2.c.s 2 35.l odd 12 2
3920.2.a.b 1 28.g odd 6 1
3920.2.a.bk 1 28.f even 6 1
4410.2.a.h 1 21.g even 6 1
4410.2.a.r 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 $$S_{2}^{\mathrm{new}}(70, [\chi])$$:

 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 4T + 16$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} - 4T + 16$$
$41$ $$(T + 7)^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} - 2T + 4$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 9T + 81$$
$71$ $$(T - 2)^{2}$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} + 10T + 100$$
$83$ $$(T + 7)^{2}$$
$89$ $$T^{2} + T + 1$$
$97$ $$(T - 14)^{2}$$