# Properties

 Label 1560.2.w.d Level $1560$ Weight $2$ Character orbit 1560.w Analytic conductor $12.457$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1560.w (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.4566627153$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} - \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{7} + (2 \beta_{2} + 2) q^{8} - q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 - b2 * q^3 + (-b3 + 2*b2 - b1) * q^4 + b2 * q^5 + (-b3 - 1) * q^6 + (-b3 + b2 - b1 + 3) * q^7 + (2*b2 + 2) * q^8 - q^9 $$q + ( - \beta_1 + 1) q^{2} - \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{7} + (2 \beta_{2} + 2) q^{8} - q^{9} + (\beta_{3} + 1) q^{10} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{3} + \beta_1) q^{12} + \beta_{2} q^{13} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{14} + q^{15} + (2 \beta_{3} - 2 \beta_1 + 4) q^{16} - 4 q^{17} + (\beta_1 - 1) q^{18} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{3} - \beta_1) q^{20} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{21} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{22} - 2 q^{23} + ( - 2 \beta_{2} + 2) q^{24} - q^{25} + (\beta_{3} + 1) q^{26} + \beta_{2} q^{27} + ( - 2 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 4) q^{28} + ( - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{29} + ( - \beta_1 + 1) q^{30} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 5) q^{31} + (4 \beta_{2} - 4 \beta_1) q^{32} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{33} + (4 \beta_1 - 4) q^{34} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{35} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{36} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{37} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{38} + q^{39} + (2 \beta_{2} - 2) q^{40} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 2) q^{41} + ( - 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{42} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{43} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{44} - \beta_{2} q^{45} + (2 \beta_1 - 2) q^{46} + (\beta_{3} - \beta_{2} + \beta_1 - 7) q^{47} + ( - 2 \beta_{3} - 2 \beta_1) q^{48} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 + 5) q^{49} + (\beta_1 - 1) q^{50} + 4 \beta_{2} q^{51} + (\beta_{3} - \beta_1) q^{52} + ( - 6 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{53} + (\beta_{3} + 1) q^{54} + (\beta_{3} - \beta_{2} + \beta_1 - 3) q^{55} + (8 \beta_{2} - 4 \beta_1 + 8) q^{56} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{57} + (2 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 6) q^{58} + ( - 5 \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{59} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{60} + (4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{61} + ( - 3 \beta_{3} + 6 \beta_{2} + 5 \beta_1 - 2) q^{62} + (\beta_{3} - \beta_{2} + \beta_1 - 3) q^{63} + 8 \beta_{2} q^{64} - q^{65} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{66} + ( - 3 \beta_{3} - 7 \beta_{2} + 3 \beta_1 - 3) q^{67} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1) q^{68} + 2 \beta_{2} q^{69} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{70} + (\beta_{3} - \beta_{2} + \beta_1 + 15) q^{71} + ( - 2 \beta_{2} - 2) q^{72} + ( - 8 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 2) q^{73} + (4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 6) q^{74} + \beta_{2} q^{75} + (2 \beta_{2} + 2 \beta_1) q^{76} + (6 \beta_{3} + 12 \beta_{2} - 6 \beta_1 + 6) q^{77} + ( - \beta_1 + 1) q^{78} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 14) q^{79} + (2 \beta_{3} + 2 \beta_1) q^{80} + q^{81} + (4 \beta_{3} - 8 \beta_{2} - 2 \beta_1 - 2) q^{82} + (\beta_{3} - 7 \beta_{2} - \beta_1 + 1) q^{83} + ( - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{84} - 4 \beta_{2} q^{85} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 4) q^{86} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 2) q^{87} + (4 \beta_{3} + 4 \beta_{2} - 4) q^{88} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 6) q^{89} + ( - \beta_{3} - 1) q^{90} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{91} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{92} + ( - 3 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 3) q^{93} + (\beta_{3} - 2 \beta_{2} + 7 \beta_1 - 8) q^{94} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{95} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{96} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{97} + ( - 6 \beta_{3} + 12 \beta_{2} - 5 \beta_1 + 11) q^{98} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 - b2 * q^3 + (-b3 + 2*b2 - b1) * q^4 + b2 * q^5 + (-b3 - 1) * q^6 + (-b3 + b2 - b1 + 3) * q^7 + (2*b2 + 2) * q^8 - q^9 + (b3 + 1) * q^10 + (b3 + 3*b2 - b1 + 1) * q^11 + (-b3 + b1) * q^12 + b2 * q^13 + (-b3 + 2*b2 - 3*b1 + 4) * q^14 + q^15 + (2*b3 - 2*b1 + 4) * q^16 - 4 * q^17 + (b1 - 1) * q^18 + (b3 - b2 - b1 + 1) * q^19 + (b3 - b1) * q^20 + (-b3 - 3*b2 + b1 - 1) * q^21 + (3*b3 + 2*b2 - b1 + 2) * q^22 - 2 * q^23 + (-2*b2 + 2) * q^24 - q^25 + (b3 + 1) * q^26 + b2 * q^27 + (-2*b3 + 6*b2 - 4*b1 + 4) * q^28 + (-4*b3 + 2*b2 + 4*b1 - 4) * q^29 + (-b1 + 1) * q^30 + (-3*b3 + 3*b2 - 3*b1 - 5) * q^31 + (4*b2 - 4*b1) * q^32 + (-b3 + b2 - b1 + 3) * q^33 + (4*b1 - 4) * q^34 + (b3 + 3*b2 - b1 + 1) * q^35 + (b3 - 2*b2 + b1) * q^36 + (-2*b3 + 4*b2 + 2*b1 - 2) * q^37 + (-b3 + 2*b2 - b1 - 2) * q^38 + q^39 + (2*b2 - 2) * q^40 + (4*b3 - 4*b2 + 4*b1 + 2) * q^41 + (-3*b3 - 2*b2 + b1 - 2) * q^42 + (-2*b3 + 2*b2 + 2*b1 - 2) * q^43 + (4*b3 + 2*b2 - 2*b1) * q^44 - b2 * q^45 + (2*b1 - 2) * q^46 + (b3 - b2 + b1 - 7) * q^47 + (-2*b3 - 2*b1) * q^48 + (-6*b3 + 6*b2 - 6*b1 + 5) * q^49 + (b1 - 1) * q^50 + 4*b2 * q^51 + (b3 - b1) * q^52 + (-6*b3 - 2*b2 + 6*b1 - 6) * q^53 + (b3 + 1) * q^54 + (b3 - b2 + b1 - 3) * q^55 + (8*b2 - 4*b1 + 8) * q^56 + (-b3 + b2 - b1 - 1) * q^57 + (2*b3 - 8*b2 + 4*b1 + 6) * q^58 + (-5*b3 + b2 + 5*b1 - 5) * q^59 + (-b3 + 2*b2 - b1) * q^60 + (4*b3 - 4*b2 - 4*b1 + 4) * q^61 + (-3*b3 + 6*b2 + 5*b1 - 2) * q^62 + (b3 - b2 + b1 - 3) * q^63 + 8*b2 * q^64 - q^65 + (-b3 + 2*b2 - 3*b1 + 4) * q^66 + (-3*b3 - 7*b2 + 3*b1 - 3) * q^67 + (4*b3 - 8*b2 + 4*b1) * q^68 + 2*b2 * q^69 + (3*b3 + 2*b2 - b1 + 2) * q^70 + (b3 - b2 + b1 + 15) * q^71 + (-2*b2 - 2) * q^72 + (-8*b3 + 8*b2 - 8*b1 + 2) * q^73 + (4*b3 - 4*b2 + 2*b1 + 6) * q^74 + b2 * q^75 + (2*b2 + 2*b1) * q^76 + (6*b3 + 12*b2 - 6*b1 + 6) * q^77 + (-b1 + 1) * q^78 + (-2*b3 + 2*b2 - 2*b1 + 14) * q^79 + (2*b3 + 2*b1) * q^80 + q^81 + (4*b3 - 8*b2 - 2*b1 - 2) * q^82 + (b3 - 7*b2 - b1 + 1) * q^83 + (-4*b3 - 2*b2 + 2*b1) * q^84 - 4*b2 * q^85 + (2*b3 - 4*b2 + 2*b1 + 4) * q^86 + (4*b3 - 4*b2 + 4*b1 + 2) * q^87 + (4*b3 + 4*b2 - 4) * q^88 + (-4*b3 + 4*b2 - 4*b1 - 6) * q^89 + (-b3 - 1) * q^90 + (b3 + 3*b2 - b1 + 1) * q^91 + (2*b3 - 4*b2 + 2*b1) * q^92 + (-3*b3 + 5*b2 + 3*b1 - 3) * q^93 + (b3 - 2*b2 + 7*b1 - 8) * q^94 + (b3 - b2 + b1 + 1) * q^95 + (-4*b3 + 4*b2) * q^96 + (-2*b3 + 2*b2 - 2*b1 - 4) * q^97 + (-6*b3 + 12*b2 - 5*b1 + 11) * q^98 + (-b3 - 3*b2 + b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{6} + 12 q^{7} + 8 q^{8} - 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^6 + 12 * q^7 + 8 * q^8 - 4 * q^9 $$4 q + 2 q^{2} - 2 q^{6} + 12 q^{7} + 8 q^{8} - 4 q^{9} + 2 q^{10} + 4 q^{12} + 12 q^{14} + 4 q^{15} + 8 q^{16} - 16 q^{17} - 2 q^{18} - 4 q^{20} - 8 q^{23} + 8 q^{24} - 4 q^{25} + 2 q^{26} + 12 q^{28} + 2 q^{30} - 20 q^{31} - 8 q^{32} + 12 q^{33} - 8 q^{34} - 8 q^{38} + 4 q^{39} - 8 q^{40} + 8 q^{41} - 12 q^{44} - 4 q^{46} - 28 q^{47} + 20 q^{49} - 2 q^{50} - 4 q^{52} + 2 q^{54} - 12 q^{55} + 24 q^{56} - 4 q^{57} + 28 q^{58} + 8 q^{62} - 12 q^{63} - 4 q^{65} + 12 q^{66} + 60 q^{71} - 8 q^{72} + 8 q^{73} + 20 q^{74} + 4 q^{76} + 2 q^{78} + 56 q^{79} + 4 q^{81} - 20 q^{82} + 12 q^{84} + 16 q^{86} + 8 q^{87} - 24 q^{88} - 24 q^{89} - 2 q^{90} - 20 q^{94} + 4 q^{95} + 8 q^{96} - 16 q^{97} + 46 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^6 + 12 * q^7 + 8 * q^8 - 4 * q^9 + 2 * q^10 + 4 * q^12 + 12 * q^14 + 4 * q^15 + 8 * q^16 - 16 * q^17 - 2 * q^18 - 4 * q^20 - 8 * q^23 + 8 * q^24 - 4 * q^25 + 2 * q^26 + 12 * q^28 + 2 * q^30 - 20 * q^31 - 8 * q^32 + 12 * q^33 - 8 * q^34 - 8 * q^38 + 4 * q^39 - 8 * q^40 + 8 * q^41 - 12 * q^44 - 4 * q^46 - 28 * q^47 + 20 * q^49 - 2 * q^50 - 4 * q^52 + 2 * q^54 - 12 * q^55 + 24 * q^56 - 4 * q^57 + 28 * q^58 + 8 * q^62 - 12 * q^63 - 4 * q^65 + 12 * q^66 + 60 * q^71 - 8 * q^72 + 8 * q^73 + 20 * q^74 + 4 * q^76 + 2 * q^78 + 56 * q^79 + 4 * q^81 - 20 * q^82 + 12 * q^84 + 16 * q^86 + 8 * q^87 - 24 * q^88 - 24 * q^89 - 2 * q^90 - 20 * q^94 + 4 * q^95 + 8 * q^96 - 16 * q^97 + 46 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2} + \zeta_{12}$$ v^2 + v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{2} + \zeta_{12}$$ -v^2 + v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( -\beta_{3} + \beta_1 ) / 2$$ (-b3 + b1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_{2}$$ b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$$.

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
781.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
−0.366025 1.36603i 1.00000i −1.73205 + 1.00000i 1.00000i −1.36603 + 0.366025i 1.26795 2.00000 + 2.00000i −1.00000 1.36603 0.366025i
781.2 −0.366025 + 1.36603i 1.00000i −1.73205 1.00000i 1.00000i −1.36603 0.366025i 1.26795 2.00000 2.00000i −1.00000 1.36603 + 0.366025i
781.3 1.36603 0.366025i 1.00000i 1.73205 1.00000i 1.00000i 0.366025 + 1.36603i 4.73205 2.00000 2.00000i −1.00000 −0.366025 1.36603i
781.4 1.36603 + 0.366025i 1.00000i 1.73205 + 1.00000i 1.00000i 0.366025 1.36603i 4.73205 2.00000 + 2.00000i −1.00000 −0.366025 + 1.36603i
 $$n$$: 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.w.d 4
4.b odd 2 1 6240.2.w.c 4
8.b even 2 1 inner 1560.2.w.d 4
8.d odd 2 1 6240.2.w.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.w.d 4 1.a even 1 1 trivial
1560.2.w.d 4 8.b even 2 1 inner
6240.2.w.c 4 4.b odd 2 1
6240.2.w.c 4 8.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 $$S_{2}^{\mathrm{new}}(1560, [\chi])$$:

 $$T_{7}^{2} - 6T_{7} + 6$$ T7^2 - 6*T7 + 6 $$T_{11}^{4} + 24T_{11}^{2} + 36$$ T11^4 + 24*T11^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T^{2} - 6 T + 6)^{2}$$
$11$ $$T^{4} + 24T^{2} + 36$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$(T + 4)^{4}$$
$19$ $$T^{4} + 8T^{2} + 4$$
$23$ $$(T + 2)^{4}$$
$29$ $$T^{4} + 104T^{2} + 1936$$
$31$ $$(T^{2} + 10 T - 2)^{2}$$
$37$ $$T^{4} + 56T^{2} + 16$$
$41$ $$(T^{2} - 4 T - 44)^{2}$$
$43$ $$T^{4} + 32T^{2} + 64$$
$47$ $$(T^{2} + 14 T + 46)^{2}$$
$53$ $$T^{4} + 224 T^{2} + 10816$$
$59$ $$T^{4} + 152T^{2} + 5476$$
$61$ $$T^{4} + 128T^{2} + 1024$$
$67$ $$T^{4} + 152T^{2} + 484$$
$71$ $$(T^{2} - 30 T + 222)^{2}$$
$73$ $$(T^{2} - 4 T - 188)^{2}$$
$79$ $$(T^{2} - 28 T + 184)^{2}$$
$83$ $$T^{4} + 104T^{2} + 2116$$
$89$ $$(T^{2} + 12 T - 12)^{2}$$
$97$ $$(T^{2} + 8 T + 4)^{2}$$