# Properties

 Label 1960.2.a.p Level $1960$ Weight $2$ Character orbit 1960.a Self dual yes Analytic conductor $15.651$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + q^{5} - 2 \beta q^{9} +O(q^{10})$$ q + (b - 1) * q^3 + q^5 - 2*b * q^9 $$q + (\beta - 1) q^{3} + q^{5} - 2 \beta q^{9} + (2 \beta - 2) q^{11} + 2 q^{13} + (\beta - 1) q^{15} + ( - 4 \beta - 2) q^{17} - 4 \beta q^{19} + (\beta - 7) q^{23} + q^{25} + ( - \beta - 1) q^{27} + ( - 2 \beta - 5) q^{29} + (2 \beta - 2) q^{31} + ( - 4 \beta + 6) q^{33} + 4 \beta q^{37} + (2 \beta - 2) q^{39} + (2 \beta + 3) q^{41} + ( - 7 \beta + 3) q^{43} - 2 \beta q^{45} + (4 \beta + 6) q^{47} + (2 \beta - 6) q^{51} - 4 \beta q^{53} + (2 \beta - 2) q^{55} + (4 \beta - 8) q^{57} - 4 q^{59} + (4 \beta + 1) q^{61} + 2 q^{65} + ( - 7 \beta - 3) q^{67} + ( - 8 \beta + 9) q^{69} - 12 q^{71} + (4 \beta - 2) q^{73} + (\beta - 1) q^{75} - 4 q^{79} + (6 \beta - 1) q^{81} + (3 \beta - 9) q^{83} + ( - 4 \beta - 2) q^{85} + ( - 3 \beta + 1) q^{87} + (4 \beta - 11) q^{89} + ( - 4 \beta + 6) q^{93} - 4 \beta q^{95} + 6 q^{97} + (4 \beta - 8) q^{99} +O(q^{100})$$ q + (b - 1) * q^3 + q^5 - 2*b * q^9 + (2*b - 2) * q^11 + 2 * q^13 + (b - 1) * q^15 + (-4*b - 2) * q^17 - 4*b * q^19 + (b - 7) * q^23 + q^25 + (-b - 1) * q^27 + (-2*b - 5) * q^29 + (2*b - 2) * q^31 + (-4*b + 6) * q^33 + 4*b * q^37 + (2*b - 2) * q^39 + (2*b + 3) * q^41 + (-7*b + 3) * q^43 - 2*b * q^45 + (4*b + 6) * q^47 + (2*b - 6) * q^51 - 4*b * q^53 + (2*b - 2) * q^55 + (4*b - 8) * q^57 - 4 * q^59 + (4*b + 1) * q^61 + 2 * q^65 + (-7*b - 3) * q^67 + (-8*b + 9) * q^69 - 12 * q^71 + (4*b - 2) * q^73 + (b - 1) * q^75 - 4 * q^79 + (6*b - 1) * q^81 + (3*b - 9) * q^83 + (-4*b - 2) * q^85 + (-3*b + 1) * q^87 + (4*b - 11) * q^89 + (-4*b + 6) * q^93 - 4*b * q^95 + 6 * q^97 + (4*b - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 $$2 q - 2 q^{3} + 2 q^{5} - 4 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} - 14 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 4 q^{31} + 12 q^{33} - 4 q^{39} + 6 q^{41} + 6 q^{43} + 12 q^{47} - 12 q^{51} - 4 q^{55} - 16 q^{57} - 8 q^{59} + 2 q^{61} + 4 q^{65} - 6 q^{67} + 18 q^{69} - 24 q^{71} - 4 q^{73} - 2 q^{75} - 8 q^{79} - 2 q^{81} - 18 q^{83} - 4 q^{85} + 2 q^{87} - 22 q^{89} + 12 q^{93} + 12 q^{97} - 16 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^17 - 14 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 4 * q^31 + 12 * q^33 - 4 * q^39 + 6 * q^41 + 6 * q^43 + 12 * q^47 - 12 * q^51 - 4 * q^55 - 16 * q^57 - 8 * q^59 + 2 * q^61 + 4 * q^65 - 6 * q^67 + 18 * q^69 - 24 * q^71 - 4 * q^73 - 2 * q^75 - 8 * q^79 - 2 * q^81 - 18 * q^83 - 4 * q^85 + 2 * q^87 - 22 * q^89 + 12 * q^93 + 12 * q^97 - 16 * 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
 −1.41421 1.41421
0 −2.41421 0 1.00000 0 0 0 2.82843 0
1.2 0 0.414214 0 1.00000 0 0 0 −2.82843 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.p 2
4.b odd 2 1 3920.2.a.bz 2
5.b even 2 1 9800.2.a.bz 2
7.b odd 2 1 1960.2.a.t 2
7.c even 3 2 280.2.q.d 4
7.d odd 6 2 1960.2.q.q 4
21.h odd 6 2 2520.2.bi.k 4
28.d even 2 1 3920.2.a.bp 2
28.g odd 6 2 560.2.q.j 4
35.c odd 2 1 9800.2.a.br 2
35.j even 6 2 1400.2.q.h 4
35.l odd 12 4 1400.2.bh.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d 4 7.c even 3 2
560.2.q.j 4 28.g odd 6 2
1400.2.q.h 4 35.j even 6 2
1400.2.bh.g 8 35.l odd 12 4
1960.2.a.p 2 1.a even 1 1 trivial
1960.2.a.t 2 7.b odd 2 1
1960.2.q.q 4 7.d odd 6 2
2520.2.bi.k 4 21.h odd 6 2
3920.2.a.bp 2 28.d even 2 1
3920.2.a.bz 2 4.b odd 2 1
9800.2.a.br 2 35.c odd 2 1
9800.2.a.bz 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1960))$$:

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{11}^{2} + 4T_{11} - 4$$ T11^2 + 4*T11 - 4 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T - 4$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 4T - 28$$
$19$ $$T^{2} - 32$$
$23$ $$T^{2} + 14T + 47$$
$29$ $$T^{2} + 10T + 17$$
$31$ $$T^{2} + 4T - 4$$
$37$ $$T^{2} - 32$$
$41$ $$T^{2} - 6T + 1$$
$43$ $$T^{2} - 6T - 89$$
$47$ $$T^{2} - 12T + 4$$
$53$ $$T^{2} - 32$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} - 2T - 31$$
$67$ $$T^{2} + 6T - 89$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 4T - 28$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 18T + 63$$
$89$ $$T^{2} + 22T + 89$$
$97$ $$(T - 6)^{2}$$