# Properties

 Label 70.2.a.a Level $70$ Weight $2$ Character orbit 70.a Self dual yes Analytic conductor $0.559$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [70,2,Mod(1,70)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(70, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("70.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$70 = 2 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 70.a (trivial)

## 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: yes Analytic conductor: $$0.558952814149$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 + q^4 - q^5 - q^7 + q^8 - 3 * q^9 $$q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - 3 q^{9} - q^{10} + 4 q^{11} - 6 q^{13} - q^{14} + q^{16} + 2 q^{17} - 3 q^{18} - q^{20} + 4 q^{22} + q^{25} - 6 q^{26} - q^{28} + 6 q^{29} + 8 q^{31} + q^{32} + 2 q^{34} + q^{35} - 3 q^{36} - 10 q^{37} - q^{40} + 2 q^{41} + 4 q^{43} + 4 q^{44} + 3 q^{45} + 8 q^{47} + q^{49} + q^{50} - 6 q^{52} - 2 q^{53} - 4 q^{55} - q^{56} + 6 q^{58} - 8 q^{59} - 14 q^{61} + 8 q^{62} + 3 q^{63} + q^{64} + 6 q^{65} - 12 q^{67} + 2 q^{68} + q^{70} - 16 q^{71} - 3 q^{72} + 2 q^{73} - 10 q^{74} - 4 q^{77} - 8 q^{79} - q^{80} + 9 q^{81} + 2 q^{82} + 8 q^{83} - 2 q^{85} + 4 q^{86} + 4 q^{88} + 10 q^{89} + 3 q^{90} + 6 q^{91} + 8 q^{94} + 2 q^{97} + q^{98} - 12 q^{99}+O(q^{100})$$ q + q^2 + q^4 - q^5 - q^7 + q^8 - 3 * q^9 - q^10 + 4 * q^11 - 6 * q^13 - q^14 + q^16 + 2 * q^17 - 3 * q^18 - q^20 + 4 * q^22 + q^25 - 6 * q^26 - q^28 + 6 * q^29 + 8 * q^31 + q^32 + 2 * q^34 + q^35 - 3 * q^36 - 10 * q^37 - q^40 + 2 * q^41 + 4 * q^43 + 4 * q^44 + 3 * q^45 + 8 * q^47 + q^49 + q^50 - 6 * q^52 - 2 * q^53 - 4 * q^55 - q^56 + 6 * q^58 - 8 * q^59 - 14 * q^61 + 8 * q^62 + 3 * q^63 + q^64 + 6 * q^65 - 12 * q^67 + 2 * q^68 + q^70 - 16 * q^71 - 3 * q^72 + 2 * q^73 - 10 * q^74 - 4 * q^77 - 8 * q^79 - q^80 + 9 * q^81 + 2 * q^82 + 8 * q^83 - 2 * q^85 + 4 * q^86 + 4 * q^88 + 10 * q^89 + 3 * q^90 + 6 * q^91 + 8 * q^94 + 2 * q^97 + q^98 - 12 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 1.00000 −1.00000 0 −1.00000 1.00000 −3.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.a.a 1
3.b odd 2 1 630.2.a.d 1
4.b odd 2 1 560.2.a.d 1
5.b even 2 1 350.2.a.b 1
5.c odd 4 2 350.2.c.b 2
7.b odd 2 1 490.2.a.h 1
7.c even 3 2 490.2.e.d 2
7.d odd 6 2 490.2.e.c 2
8.b even 2 1 2240.2.a.n 1
8.d odd 2 1 2240.2.a.q 1
11.b odd 2 1 8470.2.a.j 1
12.b even 2 1 5040.2.a.bm 1
15.d odd 2 1 3150.2.a.bj 1
15.e even 4 2 3150.2.g.c 2
20.d odd 2 1 2800.2.a.m 1
20.e even 4 2 2800.2.g.n 2
21.c even 2 1 4410.2.a.b 1
28.d even 2 1 3920.2.a.t 1
35.c odd 2 1 2450.2.a.l 1
35.f even 4 2 2450.2.c.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 1.a even 1 1 trivial
350.2.a.b 1 5.b even 2 1
350.2.c.b 2 5.c odd 4 2
490.2.a.h 1 7.b odd 2 1
490.2.e.c 2 7.d odd 6 2
490.2.e.d 2 7.c even 3 2
560.2.a.d 1 4.b odd 2 1
630.2.a.d 1 3.b odd 2 1
2240.2.a.n 1 8.b even 2 1
2240.2.a.q 1 8.d odd 2 1
2450.2.a.l 1 35.c odd 2 1
2450.2.c.k 2 35.f even 4 2
2800.2.a.m 1 20.d odd 2 1
2800.2.g.n 2 20.e even 4 2
3150.2.a.bj 1 15.d odd 2 1
3150.2.g.c 2 15.e even 4 2
3920.2.a.t 1 28.d even 2 1
4410.2.a.b 1 21.c even 2 1
5040.2.a.bm 1 12.b even 2 1
8470.2.a.j 1 11.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(70))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T - 4$$
$13$ $$T + 6$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 8$$
$37$ $$T + 10$$
$41$ $$T - 2$$
$43$ $$T - 4$$
$47$ $$T - 8$$
$53$ $$T + 2$$
$59$ $$T + 8$$
$61$ $$T + 14$$
$67$ $$T + 12$$
$71$ $$T + 16$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T - 8$$
$89$ $$T - 10$$
$97$ $$T - 2$$