# Properties

 Label 960.2.b.a Level $960$ Weight $2$ Character orbit 960.b Analytic conductor $7.666$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 960.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 + (-b2 + b1) * q^7 + (-b3 + 1) * q^9 $$q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 1) q^{9} + (\beta_{2} - \beta_1) q^{11} - \beta_{3} q^{13} - \beta_1 q^{15} - \beta_{3} q^{17} + (2 \beta_{2} + 2 \beta_1) q^{19} + ( - \beta_{3} - 2) q^{21} + ( - 3 \beta_{2} - 3 \beta_1) q^{23} + q^{25} + ( - 3 \beta_{2} + 2 \beta_1) q^{27} + 2 q^{29} + ( - 3 \beta_{2} + 3 \beta_1) q^{31} + (\beta_{3} + 2) q^{33} + (\beta_{2} - \beta_1) q^{35} + \beta_{3} q^{37} + ( - 3 \beta_{2} + \beta_1) q^{39} - 4 \beta_{3} q^{41} + ( - 3 \beta_{2} - 3 \beta_1) q^{43} + (\beta_{3} - 1) q^{45} + ( - \beta_{2} - \beta_1) q^{47} + 3 q^{49} + ( - 3 \beta_{2} + \beta_1) q^{51} + 10 q^{53} + ( - \beta_{2} + \beta_1) q^{55} + ( - 2 \beta_{3} + 8) q^{57} + (5 \beta_{2} - 5 \beta_1) q^{59} + ( - 3 \beta_{2} - \beta_1) q^{63} + \beta_{3} q^{65} + ( - \beta_{2} - \beta_1) q^{67} + (3 \beta_{3} - 12) q^{69} + (2 \beta_{2} + 2 \beta_1) q^{71} + 10 q^{73} + \beta_1 q^{75} + 4 q^{77} + (\beta_{2} - \beta_1) q^{79} + ( - 2 \beta_{3} - 7) q^{81} + ( - \beta_{2} + \beta_1) q^{83} + \beta_{3} q^{85} + 2 \beta_1 q^{87} - 2 \beta_{3} q^{89} + ( - 2 \beta_{2} - 2 \beta_1) q^{91} + ( - 3 \beta_{3} - 6) q^{93} + ( - 2 \beta_{2} - 2 \beta_1) q^{95} - 14 q^{97} + (3 \beta_{2} + \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 - q^5 + (-b2 + b1) * q^7 + (-b3 + 1) * q^9 + (b2 - b1) * q^11 - b3 * q^13 - b1 * q^15 - b3 * q^17 + (2*b2 + 2*b1) * q^19 + (-b3 - 2) * q^21 + (-3*b2 - 3*b1) * q^23 + q^25 + (-3*b2 + 2*b1) * q^27 + 2 * q^29 + (-3*b2 + 3*b1) * q^31 + (b3 + 2) * q^33 + (b2 - b1) * q^35 + b3 * q^37 + (-3*b2 + b1) * q^39 - 4*b3 * q^41 + (-3*b2 - 3*b1) * q^43 + (b3 - 1) * q^45 + (-b2 - b1) * q^47 + 3 * q^49 + (-3*b2 + b1) * q^51 + 10 * q^53 + (-b2 + b1) * q^55 + (-2*b3 + 8) * q^57 + (5*b2 - 5*b1) * q^59 + (-3*b2 - b1) * q^63 + b3 * q^65 + (-b2 - b1) * q^67 + (3*b3 - 12) * q^69 + (2*b2 + 2*b1) * q^71 + 10 * q^73 + b1 * q^75 + 4 * q^77 + (b2 - b1) * q^79 + (-2*b3 - 7) * q^81 + (-b2 + b1) * q^83 + b3 * q^85 + 2*b1 * q^87 - 2*b3 * q^89 + (-2*b2 - 2*b1) * q^91 + (-3*b3 - 6) * q^93 + (-2*b2 - 2*b1) * q^95 - 14 * q^97 + (3*b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^5 + 4 * q^9 $$4 q - 4 q^{5} + 4 q^{9} - 8 q^{21} + 4 q^{25} + 8 q^{29} + 8 q^{33} - 4 q^{45} + 12 q^{49} + 40 q^{53} + 32 q^{57} - 48 q^{69} + 40 q^{73} + 16 q^{77} - 28 q^{81} - 24 q^{93} - 56 q^{97}+O(q^{100})$$ 4 * q - 4 * q^5 + 4 * q^9 - 8 * q^21 + 4 * q^25 + 8 * q^29 + 8 * q^33 - 4 * q^45 + 12 * q^49 + 40 * q^53 + 32 * q^57 - 48 * q^69 + 40 * q^73 + 16 * q^77 - 28 * q^81 - 24 * q^93 - 56 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}$$ -v^3 - v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ -v^3 + v^2 + v $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 4$$ (b3 + b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} - \beta_{2} - \beta_1 ) / 4$$ (b3 - b2 - b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$577$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
671.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0 −1.41421 1.00000i 0 −1.00000 0 2.00000i 0 1.00000 + 2.82843i 0
671.2 0 −1.41421 + 1.00000i 0 −1.00000 0 2.00000i 0 1.00000 2.82843i 0
671.3 0 1.41421 1.00000i 0 −1.00000 0 2.00000i 0 1.00000 2.82843i 0
671.4 0 1.41421 + 1.00000i 0 −1.00000 0 2.00000i 0 1.00000 + 2.82843i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.b.a 4
3.b odd 2 1 960.2.b.b yes 4
4.b odd 2 1 inner 960.2.b.a 4
8.b even 2 1 960.2.b.b yes 4
8.d odd 2 1 960.2.b.b yes 4
12.b even 2 1 960.2.b.b yes 4
24.f even 2 1 inner 960.2.b.a 4
24.h odd 2 1 inner 960.2.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.b.a 4 1.a even 1 1 trivial
960.2.b.a 4 4.b odd 2 1 inner
960.2.b.a 4 24.f even 2 1 inner
960.2.b.a 4 24.h odd 2 1 inner
960.2.b.b yes 4 3.b odd 2 1
960.2.b.b yes 4 8.b even 2 1
960.2.b.b yes 4 8.d odd 2 1
960.2.b.b yes 4 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 $$S_{2}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{29} - 2$$ T29 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2T^{2} + 9$$
$5$ $$(T + 1)^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$(T^{2} + 8)^{2}$$
$17$ $$(T^{2} + 8)^{2}$$
$19$ $$(T^{2} - 32)^{2}$$
$23$ $$(T^{2} - 72)^{2}$$
$29$ $$(T - 2)^{4}$$
$31$ $$(T^{2} + 36)^{2}$$
$37$ $$(T^{2} + 8)^{2}$$
$41$ $$(T^{2} + 128)^{2}$$
$43$ $$(T^{2} - 72)^{2}$$
$47$ $$(T^{2} - 8)^{2}$$
$53$ $$(T - 10)^{4}$$
$59$ $$(T^{2} + 100)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} - 8)^{2}$$
$71$ $$(T^{2} - 32)^{2}$$
$73$ $$(T - 10)^{4}$$
$79$ $$(T^{2} + 4)^{2}$$
$83$ $$(T^{2} + 4)^{2}$$
$89$ $$(T^{2} + 32)^{2}$$
$97$ $$(T + 14)^{4}$$