# Properties

 Label 3960.2.a.w Level $3960$ Weight $2$ Character orbit 3960.a Self dual yes Analytic conductor $31.621$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3960.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.6207592004$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} - \beta q^{7} +O(q^{10})$$ q - q^5 - b * q^7 $$q - q^{5} - \beta q^{7} - q^{11} + 2 q^{13} + (\beta - 2) q^{17} + \beta q^{19} - 2 \beta q^{23} + q^{25} + ( - 3 \beta - 2) q^{29} + (\beta + 4) q^{31} + \beta q^{35} + ( - 3 \beta + 2) q^{37} - 2 q^{41} + 4 \beta q^{43} + (2 \beta + 8) q^{47} + (\beta - 3) q^{49} + ( - \beta - 2) q^{53} + q^{55} + ( - 2 \beta + 4) q^{59} + ( - 3 \beta + 10) q^{61} - 2 q^{65} + (4 \beta - 4) q^{67} + ( - 3 \beta + 4) q^{71} - 2 q^{73} + \beta q^{77} - 6 \beta q^{79} - 2 \beta q^{83} + ( - \beta + 2) q^{85} + (\beta + 10) q^{89} - 2 \beta q^{91} - \beta q^{95} + (2 \beta + 2) q^{97} +O(q^{100})$$ q - q^5 - b * q^7 - q^11 + 2 * q^13 + (b - 2) * q^17 + b * q^19 - 2*b * q^23 + q^25 + (-3*b - 2) * q^29 + (b + 4) * q^31 + b * q^35 + (-3*b + 2) * q^37 - 2 * q^41 + 4*b * q^43 + (2*b + 8) * q^47 + (b - 3) * q^49 + (-b - 2) * q^53 + q^55 + (-2*b + 4) * q^59 + (-3*b + 10) * q^61 - 2 * q^65 + (4*b - 4) * q^67 + (-3*b + 4) * q^71 - 2 * q^73 + b * q^77 - 6*b * q^79 - 2*b * q^83 + (-b + 2) * q^85 + (b + 10) * q^89 - 2*b * q^91 - b * q^95 + (2*b + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - q^7 $$2 q - 2 q^{5} - q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} + q^{19} - 2 q^{23} + 2 q^{25} - 7 q^{29} + 9 q^{31} + q^{35} + q^{37} - 4 q^{41} + 4 q^{43} + 18 q^{47} - 5 q^{49} - 5 q^{53} + 2 q^{55} + 6 q^{59} + 17 q^{61} - 4 q^{65} - 4 q^{67} + 5 q^{71} - 4 q^{73} + q^{77} - 6 q^{79} - 2 q^{83} + 3 q^{85} + 21 q^{89} - 2 q^{91} - q^{95} + 6 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 + q^19 - 2 * q^23 + 2 * q^25 - 7 * q^29 + 9 * q^31 + q^35 + q^37 - 4 * q^41 + 4 * q^43 + 18 * q^47 - 5 * q^49 - 5 * q^53 + 2 * q^55 + 6 * q^59 + 17 * q^61 - 4 * q^65 - 4 * q^67 + 5 * q^71 - 4 * q^73 + q^77 - 6 * q^79 - 2 * q^83 + 3 * q^85 + 21 * q^89 - 2 * q^91 - q^95 + 6 * q^97

## 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
 2.56155 −1.56155
0 0 0 −1.00000 0 −2.56155 0 0 0
1.2 0 0 0 −1.00000 0 1.56155 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## 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 3960.2.a.w 2
3.b odd 2 1 440.2.a.e 2
4.b odd 2 1 7920.2.a.bu 2
12.b even 2 1 880.2.a.o 2
15.d odd 2 1 2200.2.a.s 2
15.e even 4 2 2200.2.b.i 4
24.f even 2 1 3520.2.a.bk 2
24.h odd 2 1 3520.2.a.bp 2
33.d even 2 1 4840.2.a.j 2
60.h even 2 1 4400.2.a.bj 2
60.l odd 4 2 4400.2.b.t 4
132.d odd 2 1 9680.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.e 2 3.b odd 2 1
880.2.a.o 2 12.b even 2 1
2200.2.a.s 2 15.d odd 2 1
2200.2.b.i 4 15.e even 4 2
3520.2.a.bk 2 24.f even 2 1
3520.2.a.bp 2 24.h odd 2 1
3960.2.a.w 2 1.a even 1 1 trivial
4400.2.a.bj 2 60.h even 2 1
4400.2.b.t 4 60.l odd 4 2
4840.2.a.j 2 33.d even 2 1
7920.2.a.bu 2 4.b odd 2 1
9680.2.a.bs 2 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3960))$$:

 $$T_{7}^{2} + T_{7} - 4$$ T7^2 + T7 - 4 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} + 3T_{17} - 2$$ T17^2 + 3*T17 - 2 $$T_{19}^{2} - T_{19} - 4$$ T19^2 - T19 - 4 $$T_{23}^{2} + 2T_{23} - 16$$ T23^2 + 2*T23 - 16 $$T_{29}^{2} + 7T_{29} - 26$$ T29^2 + 7*T29 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T - 4$$
$11$ $$(T + 1)^{2}$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 3T - 2$$
$19$ $$T^{2} - T - 4$$
$23$ $$T^{2} + 2T - 16$$
$29$ $$T^{2} + 7T - 26$$
$31$ $$T^{2} - 9T + 16$$
$37$ $$T^{2} - T - 38$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} - 4T - 64$$
$47$ $$T^{2} - 18T + 64$$
$53$ $$T^{2} + 5T + 2$$
$59$ $$T^{2} - 6T - 8$$
$61$ $$T^{2} - 17T + 34$$
$67$ $$T^{2} + 4T - 64$$
$71$ $$T^{2} - 5T - 32$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + 6T - 144$$
$83$ $$T^{2} + 2T - 16$$
$89$ $$T^{2} - 21T + 106$$
$97$ $$T^{2} - 6T - 8$$