# Properties

 Label 1560.2.w.c 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: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 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.

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

Basis of coefficient ring

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

## 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
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.41421 1.00000i 2.00000 1.00000i 1.41421i −3.41421 −2.82843 −1.00000 1.41421i
781.2 −1.41421 1.00000i 2.00000 1.00000i 1.41421i −3.41421 −2.82843 −1.00000 1.41421i
781.3 1.41421 1.00000i 2.00000 1.00000i 1.41421i −0.585786 2.82843 −1.00000 1.41421i
781.4 1.41421 1.00000i 2.00000 1.00000i 1.41421i −0.585786 2.82843 −1.00000 1.41421i
 $$n$$: 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.c 4
4.b odd 2 1 6240.2.w.d 4
8.b even 2 1 inner 1560.2.w.c 4
8.d odd 2 1 6240.2.w.d 4

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

 $$T_{7}^{2} + 4T_{7} + 2$$ T7^2 + 4*T7 + 2 $$T_{11}^{4} + 24T_{11}^{2} + 16$$ T11^4 + 24*T11^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T^{2} + 4 T + 2)^{2}$$
$11$ $$T^{4} + 24T^{2} + 16$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$(T^{2} - 2)^{2}$$
$19$ $$T^{4} + 68T^{2} + 4$$
$23$ $$(T^{2} - 18)^{2}$$
$29$ $$T^{4} + 108T^{2} + 324$$
$31$ $$(T^{2} - 8 T + 8)^{2}$$
$37$ $$T^{4} + 152T^{2} + 4624$$
$41$ $$(T^{2} + 16 T + 56)^{2}$$
$43$ $$T^{4} + 24T^{2} + 16$$
$47$ $$(T^{2} + 8 T - 16)^{2}$$
$53$ $$T^{4} + 136T^{2} + 16$$
$59$ $$T^{4} + 88T^{2} + 784$$
$61$ $$(T^{2} + 64)^{2}$$
$67$ $$T^{4} + 48T^{2} + 64$$
$71$ $$(T^{2} - 4 T - 4)^{2}$$
$73$ $$(T^{2} - 20 T + 98)^{2}$$
$79$ $$(T^{2} - 8 T - 16)^{2}$$
$83$ $$T^{4} + 72T^{2} + 784$$
$89$ $$(T^{2} - 4 T - 28)^{2}$$
$97$ $$(T^{2} + 12 T - 126)^{2}$$