# Properties

 Label 1960.2.a.l Level $1960$ Weight $2$ Character orbit 1960.a Self dual yes Analytic conductor $15.651$ Analytic rank $1$ 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: $$1$$ 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 $$q + q^{3} + q^{5} - 2 q^{9} - 2 q^{11} - 4 q^{13} + q^{15} - 6 q^{19} + 3 q^{23} + q^{25} - 5 q^{27} - 3 q^{29} - 2 q^{33} - 12 q^{37} - 4 q^{39} + 7 q^{41} - 9 q^{43} - 2 q^{45} - 6 q^{53} - 2 q^{55} - 6 q^{57} + 10 q^{59} - 5 q^{61} - 4 q^{65} + 11 q^{67} + 3 q^{69} - 10 q^{71} + 8 q^{73} + q^{75} + 6 q^{79} + q^{81} + 3 q^{83} - 3 q^{87} - 17 q^{89} - 6 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^3 + q^5 - 2 * q^9 - 2 * q^11 - 4 * q^13 + q^15 - 6 * q^19 + 3 * q^23 + q^25 - 5 * q^27 - 3 * q^29 - 2 * q^33 - 12 * q^37 - 4 * q^39 + 7 * q^41 - 9 * q^43 - 2 * q^45 - 6 * q^53 - 2 * q^55 - 6 * q^57 + 10 * q^59 - 5 * q^61 - 4 * q^65 + 11 * q^67 + 3 * q^69 - 10 * q^71 + 8 * q^73 + q^75 + 6 * q^79 + q^81 + 3 * q^83 - 3 * q^87 - 17 * q^89 - 6 * q^95 + 2 * 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.l 1
4.b odd 2 1 3920.2.a.q 1
5.b even 2 1 9800.2.a.o 1
7.b odd 2 1 1960.2.a.c 1
7.c even 3 2 1960.2.q.d 2
7.d odd 6 2 280.2.q.b 2
21.g even 6 2 2520.2.bi.d 2
28.d even 2 1 3920.2.a.v 1
28.f even 6 2 560.2.q.e 2
35.c odd 2 1 9800.2.a.z 1
35.i odd 6 2 1400.2.q.c 2
35.k even 12 4 1400.2.bh.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.b 2 7.d odd 6 2
560.2.q.e 2 28.f even 6 2
1400.2.q.c 2 35.i odd 6 2
1400.2.bh.c 4 35.k even 12 4
1960.2.a.c 1 7.b odd 2 1
1960.2.a.l 1 1.a even 1 1 trivial
1960.2.q.d 2 7.c even 3 2
2520.2.bi.d 2 21.g even 6 2
3920.2.a.q 1 4.b odd 2 1
3920.2.a.v 1 28.d even 2 1
9800.2.a.o 1 5.b even 2 1
9800.2.a.z 1 35.c odd 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 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 2$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T + 6$$
$23$ $$T - 3$$
$29$ $$T + 3$$
$31$ $$T$$
$37$ $$T + 12$$
$41$ $$T - 7$$
$43$ $$T + 9$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 10$$
$61$ $$T + 5$$
$67$ $$T - 11$$
$71$ $$T + 10$$
$73$ $$T - 8$$
$79$ $$T - 6$$
$83$ $$T - 3$$
$89$ $$T + 17$$
$97$ $$T - 2$$