Properties

 Label 2960.2.a.p Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2960.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$23.6357189983$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 740) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + q^{5} + (\beta - 3) q^{7} + (2 \beta + 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + q^5 + (b - 3) * q^7 + (2*b + 1) * q^9 $$q + (\beta + 1) q^{3} + q^{5} + (\beta - 3) q^{7} + (2 \beta + 1) q^{9} + (2 \beta + 2) q^{11} + 4 q^{13} + (\beta + 1) q^{15} - 4 q^{17} + ( - 3 \beta - 1) q^{19} - 2 \beta q^{21} + (4 \beta + 2) q^{23} + q^{25} + 4 q^{27} + (2 \beta + 4) q^{29} + ( - 3 \beta - 5) q^{31} + (4 \beta + 8) q^{33} + (\beta - 3) q^{35} + q^{37} + (4 \beta + 4) q^{39} + (4 \beta - 2) q^{41} - 6 q^{43} + (2 \beta + 1) q^{45} + (\beta - 3) q^{47} + ( - 6 \beta + 5) q^{49} + ( - 4 \beta - 4) q^{51} + (4 \beta + 6) q^{53} + (2 \beta + 2) q^{55} + ( - 4 \beta - 10) q^{57} + ( - 5 \beta + 5) q^{59} + 14 q^{61} + ( - 5 \beta + 3) q^{63} + 4 q^{65} + ( - \beta + 7) q^{67} + (6 \beta + 14) q^{69} - 8 q^{71} - 6 \beta q^{73} + (\beta + 1) q^{75} - 4 \beta q^{77} + ( - 3 \beta - 1) q^{79} + ( - 2 \beta + 1) q^{81} + ( - 5 \beta + 7) q^{83} - 4 q^{85} + (6 \beta + 10) q^{87} - 2 q^{89} + (4 \beta - 12) q^{91} + ( - 8 \beta - 14) q^{93} + ( - 3 \beta - 1) q^{95} - 14 q^{97} + (6 \beta + 14) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + q^5 + (b - 3) * q^7 + (2*b + 1) * q^9 + (2*b + 2) * q^11 + 4 * q^13 + (b + 1) * q^15 - 4 * q^17 + (-3*b - 1) * q^19 - 2*b * q^21 + (4*b + 2) * q^23 + q^25 + 4 * q^27 + (2*b + 4) * q^29 + (-3*b - 5) * q^31 + (4*b + 8) * q^33 + (b - 3) * q^35 + q^37 + (4*b + 4) * q^39 + (4*b - 2) * q^41 - 6 * q^43 + (2*b + 1) * q^45 + (b - 3) * q^47 + (-6*b + 5) * q^49 + (-4*b - 4) * q^51 + (4*b + 6) * q^53 + (2*b + 2) * q^55 + (-4*b - 10) * q^57 + (-5*b + 5) * q^59 + 14 * q^61 + (-5*b + 3) * q^63 + 4 * q^65 + (-b + 7) * q^67 + (6*b + 14) * q^69 - 8 * q^71 - 6*b * q^73 + (b + 1) * q^75 - 4*b * q^77 + (-3*b - 1) * q^79 + (-2*b + 1) * q^81 + (-5*b + 7) * q^83 - 4 * q^85 + (6*b + 10) * q^87 - 2 * q^89 + (4*b - 12) * q^91 + (-8*b - 14) * q^93 + (-3*b - 1) * q^95 - 14 * q^97 + (6*b + 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} + 4 q^{11} + 8 q^{13} + 2 q^{15} - 8 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{27} + 8 q^{29} - 10 q^{31} + 16 q^{33} - 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41} - 12 q^{43} + 2 q^{45} - 6 q^{47} + 10 q^{49} - 8 q^{51} + 12 q^{53} + 4 q^{55} - 20 q^{57} + 10 q^{59} + 28 q^{61} + 6 q^{63} + 8 q^{65} + 14 q^{67} + 28 q^{69} - 16 q^{71} + 2 q^{75} - 2 q^{79} + 2 q^{81} + 14 q^{83} - 8 q^{85} + 20 q^{87} - 4 q^{89} - 24 q^{91} - 28 q^{93} - 2 q^{95} - 28 q^{97} + 28 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 + 4 * q^11 + 8 * q^13 + 2 * q^15 - 8 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^27 + 8 * q^29 - 10 * q^31 + 16 * q^33 - 6 * q^35 + 2 * q^37 + 8 * q^39 - 4 * q^41 - 12 * q^43 + 2 * q^45 - 6 * q^47 + 10 * q^49 - 8 * q^51 + 12 * q^53 + 4 * q^55 - 20 * q^57 + 10 * q^59 + 28 * q^61 + 6 * q^63 + 8 * q^65 + 14 * q^67 + 28 * q^69 - 16 * q^71 + 2 * q^75 - 2 * q^79 + 2 * q^81 + 14 * q^83 - 8 * q^85 + 20 * q^87 - 4 * q^89 - 24 * q^91 - 28 * q^93 - 2 * q^95 - 28 * q^97 + 28 * 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
 −1.73205 1.73205
0 −0.732051 0 1.00000 0 −4.73205 0 −2.46410 0
1.2 0 2.73205 0 1.00000 0 −1.26795 0 4.46410 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$-1$$

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 2960.2.a.p 2
4.b odd 2 1 740.2.a.d 2
12.b even 2 1 6660.2.a.i 2
20.d odd 2 1 3700.2.a.h 2
20.e even 4 2 3700.2.d.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.d 2 4.b odd 2 1
2960.2.a.p 2 1.a even 1 1 trivial
3700.2.a.h 2 20.d odd 2 1
3700.2.d.g 4 20.e even 4 2
6660.2.a.i 2 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3}^{2} - 2T_{3} - 2$$ T3^2 - 2*T3 - 2 $$T_{7}^{2} + 6T_{7} + 6$$ T7^2 + 6*T7 + 6 $$T_{13} - 4$$ T13 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 6T + 6$$
$11$ $$T^{2} - 4T - 8$$
$13$ $$(T - 4)^{2}$$
$17$ $$(T + 4)^{2}$$
$19$ $$T^{2} + 2T - 26$$
$23$ $$T^{2} - 4T - 44$$
$29$ $$T^{2} - 8T + 4$$
$31$ $$T^{2} + 10T - 2$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 4T - 44$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 6T + 6$$
$53$ $$T^{2} - 12T - 12$$
$59$ $$T^{2} - 10T - 50$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} - 14T + 46$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 108$$
$79$ $$T^{2} + 2T - 26$$
$83$ $$T^{2} - 14T - 26$$
$89$ $$(T + 2)^{2}$$
$97$ $$(T + 14)^{2}$$