# Properties

 Label 1960.2.a.e Level $1960$ Weight $2$ Character orbit 1960.a Self dual yes Analytic conductor $15.651$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - 2 q^{9}+O(q^{10})$$ q - q^3 + q^5 - 2 * q^9 $$q - q^{3} + q^{5} - 2 q^{9} + 2 q^{11} - q^{15} - 4 q^{17} + 2 q^{19} + q^{23} + q^{25} + 5 q^{27} + 9 q^{29} - 4 q^{31} - 2 q^{33} + 4 q^{37} - q^{41} + 9 q^{43} - 2 q^{45} + 4 q^{51} - 10 q^{53} + 2 q^{55} - 2 q^{57} + 10 q^{59} - 9 q^{61} + 5 q^{67} - q^{69} + 14 q^{71} - 12 q^{73} - q^{75} + 14 q^{79} + q^{81} - 11 q^{83} - 4 q^{85} - 9 q^{87} + 15 q^{89} + 4 q^{93} + 2 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100})$$ q - q^3 + q^5 - 2 * q^9 + 2 * q^11 - q^15 - 4 * q^17 + 2 * q^19 + q^23 + q^25 + 5 * q^27 + 9 * q^29 - 4 * q^31 - 2 * q^33 + 4 * q^37 - q^41 + 9 * q^43 - 2 * q^45 + 4 * q^51 - 10 * q^53 + 2 * q^55 - 2 * q^57 + 10 * q^59 - 9 * q^61 + 5 * q^67 - q^69 + 14 * q^71 - 12 * q^73 - q^75 + 14 * q^79 + q^81 - 11 * q^83 - 4 * q^85 - 9 * q^87 + 15 * q^89 + 4 * q^93 + 2 * q^95 + 18 * q^97 - 4 * 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.

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
0 −1.00000 0 1.00000 0 0 0 −2.00000 0
 $$n$$: 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 1960.2.a.e 1
4.b odd 2 1 3920.2.a.y 1
5.b even 2 1 9800.2.a.bc 1
7.b odd 2 1 1960.2.a.i 1
7.c even 3 2 1960.2.q.k 2
7.d odd 6 2 280.2.q.a 2
21.g even 6 2 2520.2.bi.a 2
28.d even 2 1 3920.2.a.m 1
28.f even 6 2 560.2.q.h 2
35.c odd 2 1 9800.2.a.r 1
35.i odd 6 2 1400.2.q.e 2
35.k even 12 4 1400.2.bh.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a 2 7.d odd 6 2
560.2.q.h 2 28.f even 6 2
1400.2.q.e 2 35.i odd 6 2
1400.2.bh.b 4 35.k even 12 4
1960.2.a.e 1 1.a even 1 1 trivial
1960.2.a.i 1 7.b odd 2 1
1960.2.q.k 2 7.c even 3 2
2520.2.bi.a 2 21.g even 6 2
3920.2.a.m 1 28.d even 2 1
3920.2.a.y 1 4.b odd 2 1
9800.2.a.r 1 35.c odd 2 1
9800.2.a.bc 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1960))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{11} - 2$$ T11 - 2 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T - 2$$
$13$ $$T$$
$17$ $$T + 4$$
$19$ $$T - 2$$
$23$ $$T - 1$$
$29$ $$T - 9$$
$31$ $$T + 4$$
$37$ $$T - 4$$
$41$ $$T + 1$$
$43$ $$T - 9$$
$47$ $$T$$
$53$ $$T + 10$$
$59$ $$T - 10$$
$61$ $$T + 9$$
$67$ $$T - 5$$
$71$ $$T - 14$$
$73$ $$T + 12$$
$79$ $$T - 14$$
$83$ $$T + 11$$
$89$ $$T - 15$$
$97$ $$T - 18$$