# 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

## Newspace parameters

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

## Newform invariants

 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

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

## 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.

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

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